{"id":57,"date":"2021-12-15T09:53:08","date_gmt":"2021-12-15T09:53:08","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integration-von-polynomen\/"},"modified":"2021-12-15T09:53:08","modified_gmt":"2021-12-15T09:53:08","slug":"integration-von-polynomen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integration-von-polynomen\/","title":{"raw":"Integration von Polynomen","rendered":"Integration von Polynomen"},"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: 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little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \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=\"zca313af18249\" class=\"sectionHead\"><span class=\"titlemark\">4.6 <\/span> <a id=\"x1-1220006\"><\/a>Integration von Polynomen<\/h3> <p class=\"noindent\">Wir betrachten wiederum ein Intervall <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 Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z32e131137995\"> <a id=\"x1-122001r37\"><\/a> <span class=\"ecbx-1095\">Satz 4.37 <\/span>(Riemann-Integrierbarkeit von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung einer reellen Polynomfunktion auf<\/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\">ist Riemann-integrierbar.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Monome <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">gilt<\/span> <\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <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\">Dass Polynomfunktionen eingeschr\u00e4nkt 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> Riemann-integrierbar sind, folgt, wie schon diskutiert, aus der Linearit\u00e4t des Riemann-Integrals (Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a>) und der Riemann-Integrierbarkeit von st\u00fcckweise monotonen Funktionen (Korollar <a href=\"..\/..\/chapter\/integrierbarkeit-monotoner-funktionen#x1-121004r34\">4.34<\/a>). Die zweite Aussage behandeln wir hier nur im Spezialfall <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Der Spezialfall <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist analog und die allgemeine Aussage ergibt sich aus diesen beiden Spezialf\u00e4llen und Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\">4.26<\/a>                                                                                                                                                                           (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122004r38\">4.38<\/a>). <\/p><p class=\"indent\">Da <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> monoton wachsend ist, k\u00f6nnen wir dieselbe Methode wie im Beweis von Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> (und daher auch wie in Proposition <a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\">1.1<\/a>) verwenden. Sei also <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>o<\/mi><\/math> Treppenfunktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Zerlegung in Konstanzintervalle <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <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\">wobei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/math> f\u00fcr <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> und Konstanzwert <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msubsup><\/math> respektive <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msubsup><\/math> auf <math display=\"inline\"><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><\/math> f\u00fcr <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> (siehe Beweis von Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a>). Es ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <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><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder \u00e4quivalent                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-122002r14\" \/><mstyle class=\"maketag\"><mtext>(4.14)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr gewisse Koeffizienten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/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><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit m\u00f6chten wir die linke und die rechte Summe in (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) nach unten respektive nach oben absch\u00e4tzen. Wir erhalten f\u00fcr die Summe auf der rechten Seite <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><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\">F\u00fcr die Summe auf der linken Seite von (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) erhalten wir analog                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><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\">\u2265<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/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\">\u2265<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und setzen die oben erhaltenen Ungleichungen mit (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) zusammen. Wir erhalten <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-122003r15\" \/><mstyle class=\"maketag\"><mtext>(4.15)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Aus dem Archimedischen Prinzip (Satz <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\">2.68<\/a>) folgt nun, dass <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z2001c062500f\"> <a id=\"x1-122004r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.38 <\/span>(Allgemeine Grenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen                    Sie                    Satz                    <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und dann allgemein.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen noch bemerken, dass Satz <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\">4.37<\/a> gewissermassen ein \u201ekontinuierliches Analog\u201c zu Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> darstellt. Dabei mag es aber \u00fcberraschen, dass dieses kontinuierliche Analog sogar einfacher ist. Denn in Satz <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\">4.37<\/a> tauchen im Gegensatz zu Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> keine \u201egewisse Koeffizienten <span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/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>d<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"endquote\">\u201c<\/span> auf, stattdessen gibt es eine einfache konkrete Formel. Dieses Ph\u00e4nomen, dass \u201ekontinuierliche Versionen\u201c oft einfacher sind, ist ein Grund f\u00fcr die Bedeutung der Analysis f\u00fcr die Mathematik und ebenso f\u00fcr Anwendungen der Mathematik. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf9aa46ea8091\"> <a id=\"x1-122005r39\"><\/a> <span class=\"ecbx-1095\">Applet 4.39 <\/span>(Integral eines Polynoms)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"589px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/qnq74nqf\/width\/688\/height\/589\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten  nochmals  das  partikul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re  Integral,  wobei  wir  diesmal  mit  einer<\/span> <span class=\"ecti-1095\">Polynomfunktion beginnen und dadurch Satz <\/span><a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\"><span class=\"ecti-1095\">4.37<\/span><\/a> <span class=\"ecti-1095\">anwenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5795a36d16bf\"> <a id=\"x1-122006r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 4.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Als Anwendung von Satz <\/span><a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\"><span class=\"ecti-1095\">4.37<\/span><\/a> <span class=\"ecti-1095\">berechnen wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo> <mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo> <mfrac><mrow><mn>5<\/mn><mn>2<\/mn><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/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\"><span class=\"ecti-1095\">wobei wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Definitionsbereich <\/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\">enthalten sollte, die Notation <\/span><math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z62e8ff1ae814\"> <a id=\"x1-122007r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.41 <\/span>(Integration der Wurzelfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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\">ein beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes,<\/span> <span class=\"ecti-1095\">abgeschlossenes Intervall mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie zuerst, dass <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar ist. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir des Weiteren das Riemann-Integral von<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Dazu<\/span> <span class=\"ecti-1095\">betrachten wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Zerlegung von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">aus dem Beweis von Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\"><span class=\"ecti-1095\">4.24<\/span><\/a> <span class=\"ecti-1095\">und die dort definierten Treppenfunktionen<\/span> <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das<\/span> <span class=\"ecti-1095\">Polynom <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie von <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">ausgehend eine<\/span> <span class=\"ecti-1095\">Treppenfunktion <\/span><math display=\"inline\"><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math><span class=\"ecti-1095\">respektive<\/span> <span class=\"ecti-1095\">eine Treppenfunktion <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <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> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">und berechnen Sie damit das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Betrachten Sie den Graphen von <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und spiegeln Sie ihn an der Diagonalen. Die so erhaltene Funktion ist gerade die Funktion<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter dem Graphen vor Spiegelung also durch folgendes Bild gegeben ist.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=404&amp;url=Pictures\/R-integral\/wurzelint.pdf\"><img id=\"zfcd8b17a1ef3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/wurzelint.svg\" width=\"404\"><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Versuchen Sie insbesondere bei (i) zuerst informell vorzugehen und sich an obigem Bild zu veranschaulichen, was<\/span> <span class=\"ecti-1095\">die Zuweisungen <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">sein sollten. <\/span><\/p><\/details>  <\/div> <a id=\"x1-122010r122\"><\/a> \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: 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12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \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=\"zca313af18249\" class=\"sectionHead\"><span class=\"titlemark\">4.6 <\/span> <a id=\"x1-1220006\"><\/a>Integration von Polynomen<\/h3> <p class=\"noindent\">Wir betrachten wiederum ein Intervall <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 Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z32e131137995\"> <a id=\"x1-122001r37\"><\/a> <span class=\"ecbx-1095\">Satz 4.37 <\/span>(Riemann-Integrierbarkeit von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung einer reellen Polynomfunktion auf<\/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\">ist Riemann-integrierbar.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle Monome <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u2115<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">gilt<\/span> <\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>a<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <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\">Dass Polynomfunktionen eingeschr\u00e4nkt 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> Riemann-integrierbar sind, folgt, wie schon diskutiert, aus der Linearit\u00e4t des Riemann-Integrals (Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a>) und der Riemann-Integrierbarkeit von st\u00fcckweise monotonen Funktionen (Korollar <a href=\"..\/..\/chapter\/integrierbarkeit-monotoner-funktionen#x1-121004r34\">4.34<\/a>). Die zweite Aussage behandeln wir hier nur im Spezialfall <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Der Spezialfall <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> ist analog und die allgemeine Aussage ergibt sich aus diesen beiden Spezialf\u00e4llen und Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\">4.26<\/a>                                                                                                                                                                           (siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122004r38\">4.38<\/a>). <\/p><p class=\"indent\">Da <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> monoton wachsend ist, k\u00f6nnen wir dieselbe Methode wie im Beweis von Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> (und daher auch wie in Proposition <a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\">1.1<\/a>) verwenden. Sei also <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>o<\/mi><\/math> Treppenfunktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Zerlegung in Konstanzintervalle <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <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\">wobei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/math> f\u00fcr <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> und Konstanzwert <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msubsup><\/math> respektive <math display=\"inline\"><msubsup><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msubsup><\/math> auf <math display=\"inline\"><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><\/math> f\u00fcr <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> (siehe Beweis von Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a>). Es ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <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><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>k<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mfrac><mrow><mi>b<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">oder \u00e4quivalent                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-122002r14\" \/><mstyle class=\"maketag\"><mtext>(4.14)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr gewisse Koeffizienten <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/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><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit m\u00f6chten wir die linke und die rechte Summe in (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) nach unten respektive nach oben absch\u00e4tzen. Wir erhalten f\u00fcr die Summe auf der rechten Seite <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><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\">F\u00fcr die Summe auf der linken Seite von (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) erhalten wir analog                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/munderover><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <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><msup><mrow><mi>k<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><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\">\u2265<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/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\">\u2265<\/mo> <mfrac><mrow><msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msup><mrow><mi>n<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und setzen die oben erhaltenen Ungleichungen mit (<a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122002r14\">4.14<\/a>) zusammen. Wir erhalten <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-122003r15\" \/><mstyle class=\"maketag\"><mtext>(4.15)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Aus dem Archimedischen Prinzip (Satz <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68001r68\">2.68<\/a>) folgt nun, dass <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>b<\/mi><\/mrow><mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mi>d<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac> <\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2001c062500f\"> <a id=\"x1-122004r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.38 <\/span>(Allgemeine Grenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen                    Sie                    Satz                    <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und dann allgemein.<\/span> <\/p> <\/div> <p class=\"indent\">Wir wollen noch bemerken, dass Satz <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\">4.37<\/a> gewissermassen ein \u201ekontinuierliches Analog\u201c zu Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> darstellt. Dabei mag es aber \u00fcberraschen, dass dieses kontinuierliche Analog sogar einfacher ist. Denn in Satz <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\">4.37<\/a> tauchen im Gegensatz zu Proposition <a href=\"..\/..\/chapter\/die-fakultaet-und-der-binomialsatz#x1-89001r32\">3.32<\/a> keine \u201egewisse Koeffizienten <span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>0<\/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>d<\/mi> <\/mrow> <\/msub> <\/math><\/span><span class=\"endquote\">\u201c<\/span> auf, stattdessen gibt es eine einfache konkrete Formel. Dieses Ph\u00e4nomen, dass \u201ekontinuierliche Versionen\u201c oft einfacher sind, ist ein Grund f\u00fcr die Bedeutung der Analysis f\u00fcr die Mathematik und ebenso f\u00fcr Anwendungen der Mathematik. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf9aa46ea8091\"> <a id=\"x1-122005r39\"><\/a> <span class=\"ecbx-1095\">Applet 4.39 <\/span>(Integral eines Polynoms)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"589px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/qnq74nqf\/width\/688\/height\/589\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten  nochmals  das  partikul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re  Integral,  wobei  wir  diesmal  mit  einer<\/span> <span class=\"ecti-1095\">Polynomfunktion beginnen und dadurch Satz <\/span><a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\"><span class=\"ecti-1095\">4.37<\/span><\/a> <span class=\"ecti-1095\">anwenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5795a36d16bf\"> <a id=\"x1-122006r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 4.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Als Anwendung von Satz <\/span><a href=\"..\/..\/chapter\/integration-von-polynomen#x1-122001r37\"><span class=\"ecti-1095\">4.37<\/span><\/a> <span class=\"ecti-1095\">berechnen wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup><mi>x<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo> <mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>5<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mn>5<\/mn><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><msup><mrow><mn>2<\/mn><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><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\">=<\/mo> <mfrac><mrow><mn>5<\/mn><mn>2<\/mn><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/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\"><span class=\"ecti-1095\">wobei wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Definitionsbereich <\/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\">enthalten sollte, die Notation <\/span><math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">verwendet haben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z62e8ff1ae814\"> <a id=\"x1-122007r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.41 <\/span>(Integration der Wurzelfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/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\">ein beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nktes,<\/span> <span class=\"ecti-1095\">abgeschlossenes Intervall mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie zuerst, dass <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar ist. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir des Weiteren das Riemann-Integral von<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">berechnen. Dazu<\/span> <span class=\"ecti-1095\">betrachten wir f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Zerlegung von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">aus dem Beweis von Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\"><span class=\"ecti-1095\">4.24<\/span><\/a> <span class=\"ecti-1095\">und die dort definierten Treppenfunktionen<\/span> <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r das<\/span> <span class=\"ecti-1095\">Polynom <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie von <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">ausgehend eine<\/span> <span class=\"ecti-1095\">Treppenfunktion <\/span><math display=\"inline\"><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math><span class=\"ecti-1095\">respektive<\/span> <span class=\"ecti-1095\">eine Treppenfunktion <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <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> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><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> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">und berechnen Sie damit das Integral <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Betrachten Sie den Graphen von <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und spiegeln Sie ihn an der Diagonalen. Die so erhaltene Funktion ist gerade die Funktion<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac> <\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">deren Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che unter dem Graphen vor Spiegelung also durch folgendes Bild gegeben ist.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=404&amp;url=Pictures\/R-integral\/wurzelint.pdf\"><img decoding=\"async\" id=\"zfcd8b17a1ef3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/wurzelint.svg\" width=\"404\" \/><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Versuchen Sie insbesondere bei (i) zuerst informell vorzugehen und sich an obigem Bild zu veranschaulichen, was<\/span> <span class=\"ecti-1095\">die Zuweisungen <\/span><math display=\"inline\"><mi>u<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">sein sollten. <\/span><\/p><\/details>  <\/div> <a id=\"x1-122010r122\"><\/a> \n","protected":false},"author":1089,"menu_order":6,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-57","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/57","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\/57\/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\/57\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=57"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=57"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}