{"id":55,"date":"2021-12-15T09:53:08","date_gmt":"2021-12-15T09:53:08","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/anwendungen\/"},"modified":"2021-12-15T09:53:08","modified_gmt":"2021-12-15T09:53:08","slug":"anwendungen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/anwendungen\/","title":{"raw":"Anwendungen","rendered":"Anwendungen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style>\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z062002c7cbe7\" class=\"sectionHead\"><span class=\"titlemark\">4.4 <\/span> <a id=\"x1-1150004\"><\/a>Anwendungen<\/h3> <a id=\"x1-115001r114\"><\/a> <h4 id=\"z7eaf56922d29\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.1 <\/span> <a id=\"x1-1160001\"><\/a>Intervallfunktionen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun spezielle Abbildungen auf der Menge der Teilintervalle eines Intervalles betrachten, wobei wir Ordnungsvertauschungen im Stile von (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114007r9\">4.9<\/a>) zulassen wollen. Genauer untersuchen wir folgenden Begriff. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zae4dc480ee56\"> <a id=\"x1-116001r29\"><\/a> <span class=\"ecbx-1095\">Definition 4.29.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und sei <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Wir nennen <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> eine <span class=\"ecbx-1095\">additive<\/span> <span class=\"ecbx-1095\">Intervallfunktion <\/span>auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> (i)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> (ii)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> (iii)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Wir wollen hier kurz erkl\u00e4ren, woher die Bezeichnung \u201eadditive Intervallfunktion\u201c stammt. Ist <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> eine additive Intervallfunktion auf einem kompakten Intervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> so kann man eine reellwertige Funktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <\/math> auf der Menge der nicht-leeren Teilintervalle von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren. Diese hat die Eigenschaften                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116005r11\" \/><mstyle class=\"maketag\"><mtext>(4.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b3<\/mi><\/math> in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (wieso?). Vor allem letztere Eigenschaft begr\u00fcndet die Bezeichnung \u201eadditive Intervallfunktion\u201c. <\/p><p class=\"indent\">Hat man umgekehrt eine reellwertige Funktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <\/math> auf der Menge der nicht-leeren Teilintervalle von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gegeben, die (<a href=\"..\/..\/chapter\/anwendungen#x1-116005r11\">4.11<\/a>)gen\u00fcgt, so definiert <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b2<\/mi><\/math> und <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>\u03b2<\/mi><\/math> eine additive Intervallfunktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (wieso?). <\/p><p class=\"indent\">Somit haben wir also zwei Arten, wie wir uns additive Intervallfunktionen vorstellen k\u00f6nnen. Eine grosse Kollektion von Beispielen erh\u00e4lt man mit Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\">4.26<\/a> und \u00dcbung <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114008r27\">4.27<\/a>, nach welchen die Abbildung <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2110<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116006r12\" \/><mstyle class=\"maketag\"><mtext>(4.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr jede Riemann-integrierbare Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine additive Intervallfunktion ist. Die folgende Proposition charakterisiert derartige additive Intervallfunktionen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zcade9410f73c\"> <a id=\"x1-116007r30\"><\/a> <span class=\"ecbx-1095\">Proposition 4.30.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare<\/span> <span class=\"ecti-1095\">Funktion und <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math><span class=\"ecti-1095\">eine additive<\/span> <span class=\"ecti-1095\">Intervallfunktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116008r13\" \/><mstyle class=\"maketag\"><mtext>(4.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir m\u00f6chten anmerken, dass jedoch nicht alle additiven Intervallfunktionen von der Form in (<a href=\"..\/..\/chapter\/anwendungen#x1-116006r12\">4.12<\/a>)sein m\u00fcssen. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> eine Treppenfunktion auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Zerlegung <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> in Konstanzintervalle von <span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span> Seien <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> die Konstanzwerte von <math display=\"inline\"><mi>u<\/mi><\/math> bez\u00fcglich <span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auf Grund der Annahme <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> folgt <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Unter Verwendung der Additivit\u00e4t von <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> erh\u00e4lt man damit f\u00fcr die Untersumme <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ebenso ergibt sich <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fcr jede Treppenfunktion <math display=\"inline\"><mi>o<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher gelten f\u00fcr das untere Integral <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><\/math> und das obere Integral <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> von <math display=\"inline\"><mi>f<\/mi><\/math> \u00fcber <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Ungleichungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mo class=\"MathClass-rel\">\u2264<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>f<\/mi><\/math> aber Riemann-integrierbar ist, gilt <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> und somit <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Man kann Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> als Wegweiser verwenden, um verschiedene Interpretationen des Riemann-Integrals zu finden. Formal gesehen sind diese Anwendungen jeweils Definitionen. <a id=\"x1-116009r116\"><\/a> <\/p> <h4 id=\"zcfaece744d77\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.2 <\/span> <a id=\"x1-1170002\"><\/a>Fl\u00e4cheninhalt<\/h4> <p class=\"noindent\">Die einfachste Anwendung von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> ist die Interpretation von <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als <span class=\"ecbx-1095\">Fl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">cheninhalt des Gebietes<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">unter dem Graphen einer Riemann-integrierbaren Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Argumentation, die zu dieser Definition f\u00fchrt, haben wir bereits in Abschnitt <a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-40001\">1.1<\/a> besprochen. Formal gesehen erachten wir <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als Definition des Fl\u00e4cheninhalts des obigen Gebietes. <a id=\"x1-117001r117\"><\/a> <\/p> <h4 id=\"za4b4399e7dc2\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.3 <\/span> <a id=\"x1-1180003\"><\/a>Masse, Momente und Schwerpunkt<\/h4> <p class=\"noindent\">Es gibt nat\u00fcrlich auch viele physikalische Beispiele f\u00fcr die Bedeutung des Riemann-Integrals. Sei zum Beispiel <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und sei&nbsp;<math display=\"inline\"><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Dichte eines Stabes (in Kilogramm pro Meter,&nbsp;<math display=\"inline\"><mfrac><mrow><mi>k<\/mi><mi>g<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/math>) bei der Koordinate <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann ergibt sich aus Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a>, dass wir <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als das <span class=\"ecbx-1095\">Gesamtmasse <\/span>(in <math display=\"inline\"><mi>k<\/mi><mi>g<\/mi><\/math>) interpretieren sollten. (Wieso?) <\/p><p class=\"indent\">Wir erinnern daran, dass bei einem Hebel das Moment (in <math display=\"inline\"><mi>N<\/mi><mi>m<\/mi><\/math>) einer Krafteinwirkung durch das Produkt der Krafteinwirkung (in Newton <math display=\"inline\"><mi>N<\/mi><\/math>) und des Weges (in <math display=\"inline\"><mi>m<\/mi><\/math>) definiert ist. Wir stellen uns vor, dass <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> der obige Stab mit Dichtefunktion <math display=\"inline\"><mi>\u03c1<\/mi><\/math> im Ursprung fixiert ist, und die Schwerkraft (mit Gravitationskonstante <math display=\"inline\"><mi>g<\/mi><\/math> in <math display=\"inline\"><mi>N<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>k<\/mi><mi>g<\/mi><\/math>) auf den Stab einwirkt. In diesem Fall ergibt sich f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass die dem Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> entsprechende Masse <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Ungleichung <\/p><table id=\"z6995117e3b17\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>m<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">erf\u00fcllt, woraus sich f\u00fcr das entsprechende Moment <math display=\"inline\"><mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Ungleichung <\/p> <table id=\"zbb3adf735b62\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>g<\/mi><mi>\u03b1<\/mi><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>g<\/mi><mi>\u03b2<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">ergibt. Diese Eigenschaft von <math display=\"inline\"><mi>M<\/mi><\/math> unterscheidet sich zwar formal von (<a href=\"..\/..\/chapter\/anwendungen#x1-116008r13\">4.13<\/a>) doch l\u00e4sst sich mit Hilfe der Stetigkeit von <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>x<\/mi><\/math> der Beweis von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> anpassen. Ebenso ist es physikalisch sinnvoll die Additivit\u00e4t dieser Momentfunktion anzunehmen, dadurch erhalten wir die Definition <\/p><table id=\"z06b17ef23bcb\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>M<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>g<\/mi><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr das <span class=\"ecbx-1095\">Gesamtmoment <\/span>des Stabes. <\/p><p class=\"indent\">Der <span class=\"ecbx-1095\">Schwerpunkt <\/span>des Stabes ist definiert als die <math display=\"inline\"><mi>x<\/mi><\/math>-Koordinate <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> so dass eine Punktmasse bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> mit derselben Masse wie der Stab auch dasselbe Moment besitzt. Also ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>m<\/mi><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der Schwerpunkt des Stabes. <\/p><p class=\"indent\">Die Annahme <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist f\u00fcr diese Diskussion (abgesehen von der Vorstellung dass der Stab am Ursprung gehalten wird) nicht notwendig, falls <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> erhalten wir physikalisch sinnvolle Integrale von Funktionen, die sowohl positive als auch negative Werte annehmen k\u00f6nnen. <a id=\"x1-118001r118\"><\/a> <\/p> <h4 id=\"z5c5044b0fcee\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.4 <\/span> <a id=\"x1-1190004\"><\/a>Geleistete Arbeit<\/h4> <p class=\"noindent\">Wenn <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> ist und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Riemann-integrierbare Funktion ist, die zu einem Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> den Energieverbrauch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (in Watt <math display=\"inline\"><mi>W<\/mi><\/math>) zum Beispiel in Form elektrischer Energie eines Hauses angibt, so beschreibt <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> die verbrauchte <span class=\"ecbx-1095\">Energie <\/span>oder vom Stromnetz eingespeiste <span class=\"ecbx-1095\">Arbeit <\/span>(in Joule <math display=\"inline\"><mi>J<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>W<\/mi><mi>s<\/mi><\/math>) zwischen den Zeitpunkten <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math> (in Sekunden <math display=\"inline\"><mi>s<\/mi><\/math>). Diese Interpretation ergibt sich wiederum aus Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> und der Definition, dass Arbeit gleich Leistung mal Zeitdauer ist. Hier ist es ebenso physikalisch sinnvoll, Funktionen mit positiven und negativen Werten zuzulassen, wenn zum Beispiel das Hausdach mit einer Solaranlage ausgestattet ist, die bei Sch\u00f6nwetter etwaige Energie\u00fcbersch\u00fcsse des Hauses ins Stromnetz zur\u00fcckspeist. Das Vorzeichen des Integrals entscheidet in diesem Fall, ob insgesamt innerhalb der Zeitspanne <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> das Haus ein Energieverbraucher oder Energielieferant war. <a id=\"x1-119001r119\"><\/a> <\/p> <h4 id=\"z6e1480301927\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.5 <\/span> <a id=\"x1-1200005\"><\/a>Vorteil des Integralbegriffs<\/h4> <p class=\"noindent\">Wir haben das Integral abstrakt mittels der Definition <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109005r6\">4.6<\/a> des Integrals einer Treppenfunktion und der Definition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110003r11\">4.11<\/a> des Integrals einer Riemann-integrierbaren Funktion eingef\u00fchrt. Bei Besprechung dieser Definitionen haben wir uns zwar von einer geometrischen Interpretation des Integrals als (vorzeichenbehafteter) Fl\u00e4cheninhalt leiten lassen, doch war diese Vorstellung formal nicht notwendig f\u00fcr unsere Diskussionen. Wir hoffen, dass der Vorteil dieses abstrakten Zugangs nun ersichtlich ist: Das Integral hat je nach Zusammenhang verschiedene (zum Beispiel physikalische) Bedeutungen. Wenn unsere Definition des Integrals \u201eder Fl\u00e4cheninhalt unter der Kurve\u201c gewesen w\u00e4re, dann w\u00e4re es nicht klar, was genau der Zusammenhang zwischen einem Fl\u00e4cheninhalt und einer Momentberechnung sein sollte.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Des Weiteren w\u00e4re diese Definition zirkul\u00e4r gewesen, da wir ohne Definition des Integrals keine Definition des Fl\u00e4cheninhalts unter der Kurve haben.<\/span><\/span> In diesem Sinne ist unser abstrakter Zugang nicht Selbstzweck, sondern geradezu notwendig auf Grund der vielf\u00e4ltigen Anwendungen des Integralbegriffs.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-120001r115\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z062002c7cbe7\" class=\"sectionHead\"><span class=\"titlemark\">4.4 <\/span> <a id=\"x1-1150004\"><\/a>Anwendungen<\/h3> <a id=\"x1-115001r114\"><\/a> <h4 id=\"z7eaf56922d29\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.1 <\/span> <a id=\"x1-1160001\"><\/a>Intervallfunktionen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun spezielle Abbildungen auf der Menge der Teilintervalle eines Intervalles betrachten, wobei wir Ordnungsvertauschungen im Stile von (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114007r9\">4.9<\/a>) zulassen wollen. Genauer untersuchen wir folgenden Begriff. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zae4dc480ee56\"> <a id=\"x1-116001r29\"><\/a> <span class=\"ecbx-1095\">Definition 4.29.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mi>\u211d<\/mi><\/math> und sei <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Wir nennen <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> eine <span class=\"ecbx-1095\">additive<\/span> <span class=\"ecbx-1095\">Intervallfunktion <\/span>auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> falls <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> (i)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> (ii)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> (iii)<\/dt><dd class=\"enumerate\">F\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Wir wollen hier kurz erkl\u00e4ren, woher die Bezeichnung \u201eadditive Intervallfunktion\u201c stammt. Ist <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> eine additive Intervallfunktion auf einem kompakten Intervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> so kann man eine reellwertige Funktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <\/math> auf der Menge der nicht-leeren Teilintervalle von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> durch <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren. Diese hat die Eigenschaften                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u222a<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116005r11\" \/><mstyle class=\"maketag\"><mtext>(4.11)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b3<\/mi><\/math> in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (wieso?). Vor allem letztere Eigenschaft begr\u00fcndet die Bezeichnung \u201eadditive Intervallfunktion\u201c. <\/p><p class=\"indent\">Hat man umgekehrt eine reellwertige Funktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <\/math> auf der Menge der nicht-leeren Teilintervalle von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gegeben, die (<a href=\"..\/..\/chapter\/anwendungen#x1-116005r11\">4.11<\/a>)gen\u00fcgt, so definiert <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>\u03b2<\/mi><\/math> und <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi mathvariant=\"bold-script\">\ud835\udca5<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>\u03b2<\/mi><\/math> eine additive Intervallfunktion <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (wieso?). <\/p><p class=\"indent\">Somit haben wir also zwei Arten, wie wir uns additive Intervallfunktionen vorstellen k\u00f6nnen. Eine grosse Kollektion von Beispielen erh\u00e4lt man mit Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\">4.26<\/a> und \u00dcbung <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114008r27\">4.27<\/a>, nach welchen die Abbildung <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2110<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116006r12\" \/><mstyle class=\"maketag\"><mtext>(4.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr jede Riemann-integrierbare Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine additive Intervallfunktion ist. Die folgende Proposition charakterisiert derartige additive Intervallfunktionen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zcade9410f73c\"> <a id=\"x1-116007r30\"><\/a> <span class=\"ecbx-1095\">Proposition 4.30.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare<\/span> <span class=\"ecti-1095\">Funktion und <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math><span class=\"ecti-1095\">eine additive<\/span> <span class=\"ecti-1095\">Intervallfunktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-116008r13\" \/><mstyle class=\"maketag\"><mtext>(4.13)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann<\/span> <span class=\"ecti-1095\">ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir m\u00f6chten anmerken, dass jedoch nicht alle additiven Intervallfunktionen von der Form in (<a href=\"..\/..\/chapter\/anwendungen#x1-116006r12\">4.12<\/a>)sein m\u00fcssen. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> eine Treppenfunktion auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Zerlegung <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> in Konstanzintervalle von <span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi><\/math><\/span><span class=\"period\">.<\/span> Seien <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> die Konstanzwerte von <math display=\"inline\"><mi>u<\/mi><\/math> bez\u00fcglich <span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auf Grund der Annahme <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> folgt <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Unter Verwendung der Additivit\u00e4t von <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><\/math> erh\u00e4lt man damit f\u00fcr die Untersumme <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\u2110<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ebenso ergibt sich <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> f\u00fcr jede Treppenfunktion <math display=\"inline\"><mi>o<\/mi><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher gelten f\u00fcr das untere Integral <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><\/math> und das obere Integral <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> von <math display=\"inline\"><mi>f<\/mi><\/math> \u00fcber <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Ungleichungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mo class=\"MathClass-rel\">\u2264<\/mo><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>f<\/mi><\/math> aber Riemann-integrierbar ist, gilt <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> und somit <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2110<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u03b2<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Man kann Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> als Wegweiser verwenden, um verschiedene Interpretationen des Riemann-Integrals zu finden. Formal gesehen sind diese Anwendungen jeweils Definitionen. <a id=\"x1-116009r116\"><\/a> <\/p> <h4 id=\"zcfaece744d77\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.2 <\/span> <a id=\"x1-1170002\"><\/a>Fl\u00e4cheninhalt<\/h4> <p class=\"noindent\">Die einfachste Anwendung von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> ist die Interpretation von <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als <span class=\"ecbx-1095\">Fl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">cheninhalt des Gebietes<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">unter dem Graphen einer Riemann-integrierbaren Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Die Argumentation, die zu dieser Definition f\u00fchrt, haben wir bereits in Abschnitt <a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-40001\">1.1<\/a> besprochen. Formal gesehen erachten wir <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als Definition des Fl\u00e4cheninhalts des obigen Gebietes. <a id=\"x1-117001r117\"><\/a> <\/p> <h4 id=\"za4b4399e7dc2\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.3 <\/span> <a id=\"x1-1180003\"><\/a>Masse, Momente und Schwerpunkt<\/h4> <p class=\"noindent\">Es gibt nat\u00fcrlich auch viele physikalische Beispiele f\u00fcr die Bedeutung des Riemann-Integrals. Sei zum Beispiel <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und sei&nbsp;<math display=\"inline\"><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Dichte eines Stabes (in Kilogramm pro Meter,&nbsp;<math display=\"inline\"><mfrac><mrow><mi>k<\/mi><mi>g<\/mi><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/math>) bei der Koordinate <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann ergibt sich aus Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a>, dass wir <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> als das <span class=\"ecbx-1095\">Gesamtmasse <\/span>(in <math display=\"inline\"><mi>k<\/mi><mi>g<\/mi><\/math>) interpretieren sollten. (Wieso?) <\/p><p class=\"indent\">Wir erinnern daran, dass bei einem Hebel das Moment (in <math display=\"inline\"><mi>N<\/mi><mi>m<\/mi><\/math>) einer Krafteinwirkung durch das Produkt der Krafteinwirkung (in Newton <math display=\"inline\"><mi>N<\/mi><\/math>) und des Weges (in <math display=\"inline\"><mi>m<\/mi><\/math>) definiert ist. Wir stellen uns vor, dass <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> der obige Stab mit Dichtefunktion <math display=\"inline\"><mi>\u03c1<\/mi><\/math> im Ursprung fixiert ist, und die Schwerkraft (mit Gravitationskonstante <math display=\"inline\"><mi>g<\/mi><\/math> in <math display=\"inline\"><mi>N<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>k<\/mi><mi>g<\/mi><\/math>) auf den Stab einwirkt. In diesem Fall ergibt sich f\u00fcr <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass die dem Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> entsprechende Masse <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Ungleichung <\/p><table id=\"z6995117e3b17\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>m<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">erf\u00fcllt, woraus sich f\u00fcr das entsprechende Moment <math display=\"inline\"><mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die Ungleichung <\/p> <table id=\"zbb3adf735b62\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>g<\/mi><mi>\u03b1<\/mi><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>g<\/mi><mi>\u03b2<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>\u03c1<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\">ergibt. Diese Eigenschaft von <math display=\"inline\"><mi>M<\/mi><\/math> unterscheidet sich zwar formal von (<a href=\"..\/..\/chapter\/anwendungen#x1-116008r13\">4.13<\/a>) doch l\u00e4sst sich mit Hilfe der Stetigkeit von <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>x<\/mi><\/math> der Beweis von Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> anpassen. Ebenso ist es physikalisch sinnvoll die Additivit\u00e4t dieser Momentfunktion anzunehmen, dadurch erhalten wir die Definition <\/p><table id=\"z06b17ef23bcb\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>M<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>g<\/mi><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <\/math><\/td><\/tr><\/table> <p class=\"indent\">f\u00fcr das <span class=\"ecbx-1095\">Gesamtmoment <\/span>des Stabes. <\/p><p class=\"indent\">Der <span class=\"ecbx-1095\">Schwerpunkt <\/span>des Stabes ist definiert als die <math display=\"inline\"><mi>x<\/mi><\/math>-Koordinate <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> so dass eine Punktmasse bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> mit derselben Masse wie der Stab auch dasselbe Moment besitzt. Also ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>M<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>m<\/mi><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>\u03c1<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der Schwerpunkt des Stabes. <\/p><p class=\"indent\">Die Annahme <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist f\u00fcr diese Diskussion (abgesehen von der Vorstellung dass der Stab am Ursprung gehalten wird) nicht notwendig, falls <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> erhalten wir physikalisch sinnvolle Integrale von Funktionen, die sowohl positive als auch negative Werte annehmen k\u00f6nnen. <a id=\"x1-118001r118\"><\/a> <\/p> <h4 id=\"z5c5044b0fcee\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.4 <\/span> <a id=\"x1-1190004\"><\/a>Geleistete Arbeit<\/h4> <p class=\"noindent\">Wenn <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> ist und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Riemann-integrierbare Funktion ist, die zu einem Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> den Energieverbrauch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (in Watt <math display=\"inline\"><mi>W<\/mi><\/math>) zum Beispiel in Form elektrischer Energie eines Hauses angibt, so beschreibt <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> die verbrauchte <span class=\"ecbx-1095\">Energie <\/span>oder vom Stromnetz eingespeiste <span class=\"ecbx-1095\">Arbeit <\/span>(in Joule <math display=\"inline\"><mi>J<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>W<\/mi><mi>s<\/mi><\/math>) zwischen den Zeitpunkten <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> und <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math> (in Sekunden <math display=\"inline\"><mi>s<\/mi><\/math>). Diese Interpretation ergibt sich wiederum aus Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> und der Definition, dass Arbeit gleich Leistung mal Zeitdauer ist. Hier ist es ebenso physikalisch sinnvoll, Funktionen mit positiven und negativen Werten zuzulassen, wenn zum Beispiel das Hausdach mit einer Solaranlage ausgestattet ist, die bei Sch\u00f6nwetter etwaige Energie\u00fcbersch\u00fcsse des Hauses ins Stromnetz zur\u00fcckspeist. Das Vorzeichen des Integrals entscheidet in diesem Fall, ob insgesamt innerhalb der Zeitspanne <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> das Haus ein Energieverbraucher oder Energielieferant war. <a id=\"x1-119001r119\"><\/a> <\/p> <h4 id=\"z6e1480301927\" class=\"subsectionHead\"><span class=\"titlemark\">4.4.5 <\/span> <a id=\"x1-1200005\"><\/a>Vorteil des Integralbegriffs<\/h4> <p class=\"noindent\">Wir haben das Integral abstrakt mittels der Definition <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109005r6\">4.6<\/a> des Integrals einer Treppenfunktion und der Definition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110003r11\">4.11<\/a> des Integrals einer Riemann-integrierbaren Funktion eingef\u00fchrt. Bei Besprechung dieser Definitionen haben wir uns zwar von einer geometrischen Interpretation des Integrals als (vorzeichenbehafteter) Fl\u00e4cheninhalt leiten lassen, doch war diese Vorstellung formal nicht notwendig f\u00fcr unsere Diskussionen. Wir hoffen, dass der Vorteil dieses abstrakten Zugangs nun ersichtlich ist: Das Integral hat je nach Zusammenhang verschiedene (zum Beispiel physikalische) Bedeutungen. Wenn unsere Definition des Integrals \u201eder Fl\u00e4cheninhalt unter der Kurve\u201c gewesen w\u00e4re, dann w\u00e4re es nicht klar, was genau der Zusammenhang zwischen einem Fl\u00e4cheninhalt und einer Momentberechnung sein sollte.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Des Weiteren w\u00e4re diese Definition zirkul\u00e4r gewesen, da wir ohne Definition des Integrals keine Definition des Fl\u00e4cheninhalts unter der Kurve haben.<\/span><\/span> In diesem Sinne ist unser abstrakter Zugang nicht Selbstzweck, sondern geradezu notwendig auf Grund der vielf\u00e4ltigen Anwendungen des Integralbegriffs.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-120001r115\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":4,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-55","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/55","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/55\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/55\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=55"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=55"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=55"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}