Author: Werner Knobloch
Date: 23 März 2013

Einfache bestimmte Integrale über Potenzfunktionen

Werner Knobloch, 2013

explizite Berechnung als Grenzwert von Unter- bzw. Obersumme

$ \int_{0}^{b}{\textcolor{#800000}p \cdot  x^\textcolor{#009900}k dx  } =\textcolor{#800000}p \cdot  \frac{1}{\textcolor{#009900}k+1}\cdot  b^ {\textcolor{#009900}k +1}

U(n)= \sum_{j=0}^{n-1}{Δx \cdot f(j\cdot Δx )= \sum_{j=0}^{n-1}{Δx \cdot p \cdot (j\cdot Δx )^k =p  \cdot  Δx\cdot  \sum_{j=0}^{n-1}{ j^k }=p  \cdot  \frac{b^{k+1} }{n^{k+1}}\cdot   \sum_{j=0}^{n-1}{ j^k }= p  \cdot  \frac{b^{k+1}  }{n^{k+1}  }\cdot \left(   \frac{1} {k+1  }   {n^{k+1} } + \textcolor{#00ffff}{O_1(n^k)}    \right) } = p  \cdot  b^{k+1}   \cdot  \left(   \frac {1} {k+1}    +\frac { \textcolor{#00ffff}{O_1(n^k)}} {n^{k+1} }   \right) }  

O(n)= \sum_{j=10}^{n}{Δx \cdot f(j\cdot Δx )= \sum_{j=1}^{n}{Δx \cdot p \cdot (j\cdot Δx )^k =p  \cdot  Δx^  \cdot  \sum_{j=1}^{n}{ j^k }=p  \cdot  \frac{b^{k+1} }{n^{k+1}}\cdot   \sum_{j=1}^{n }{ j^ k }= p  \cdot  \frac{b^{k+1}  }{n^{k+1}  }\cdot \left(   \frac{1} {k+1  }   {n^{k+1} } +  \textcolor{#00ffff}{O_2(n^k) }   \right) } = p  \cdot  b^{k+1}   \cdot  \left(   \frac {1} {k+1}    +\frac {  \textcolor{#00ffff}{O_2(n^k)}}  {n^{k+1} }   \right) }  

Formeln für den allgemeinen Fall

\int_{0}^{b}{ f(x)dx }=  \lim_{ n \to \infty }U(n)  =    \lim_{ n \to \infty } p  \cdot b^{k+1}   \cdot  \left(   \frac {1} {k+1}    +\frac {  \textcolor{#00ffff}{O_1(n^k)}}  {n^{k+1}}    \right)           = p  \cdot  b^{k+1}   \cdot   \lim_{ n \to \infty }  \left(   \frac {1} {k+1}    +\frac { \textcolor{#00ffff}{O_1(n^k)}}  {n^{k+1}}    \right)  = p  \cdot b^{k+1}   \cdot    \frac {1} {k+1}   = p    \cdot    \frac {1} {k+1} \cdot b^{k+1}  

\int_{0}^{b}{ f(x)dx }=  \lim_{ n \to \infty }O(n)  =    \lim_{ n \to \infty } p  \cdot b^{k+1}   \cdot  \left(   \frac {1} {k+1}    +\frac { \textcolor{#00ffff}{O_2(n^k)}}  {n^{k+1}}    \right)           = p  \cdot  b^{k+1}   \cdot   \lim_{ n \to \infty }  \left(   \frac {1} {k+1}    +\frac { \textcolor{#00ffff}{O_2(n^k)}}  {n^{k+1}}    \right)  = p  \cdot b^{k+1}   \cdot    \frac {1} {k+1}   = p    \cdot    \frac {1} {k+1} \cdot b^{k+1}  

 \text{Die Terme   } \textcolor{#00ffff}{ O_1(n^k) }     \text{ und   } \textcolor{#00ffff}{O_2(n^k) } \text{ enthalten nur Potenzen vom Grad ≤ k} \
\text{Faulhabersche Formel (Wikipedia)}

n = 9 (slider from 1 to 800, step 1)
r_5(n) = n^(-6)
r_4(n) = n^(-5)
r_3(n) = n^(-4)
r_2(n) = n^(-3)
r_1(n) = n^(-2)
r_0(n) = n^(-1)
Liste7: {r_0, r_1, r_2, r_3, r_4, r_5}
k_5(n) = 1 - n^(5)
k_4(n) = 1 - n^(4)
k_3(n) = 1 - n^(3)
k_2(n) = 1 - n^(2)
k_1(n) = 1 - n^(1)
k_0(n) = 1 - n^(0)
UListe5: {k_0, k_1, k_2, k_3, k_4, k_5}
s_5(n) = (1 / 6 * n^(6)) + (1 / 2 * n^(5)) + (5 / 12 * n^(4)) - (1 / 12 * n^(2))
s_4(n) = (1 / 5 * n^(5)) + (1 / 2 * n^(4)) + (1 / 3 * n^(3)) - (1 / 30 * n)
s_3(n) = (1 / 4 * n^(4)) + (1 / 2 * n^(3)) + (1 / 4 * n^(2))
s_2(n) = (1 / 3 * n^(3)) + (1 / 2 * n^(2)) + (1 / 6 * n)
s_1(n) = (1 / 2 * n^(2)) + (1 / 2 * n)
s_0(n) = n
OListe4: {s_0, s_1, s_2, s_3, s_4, s_5}
UListe6 = Sequence(Simplify[Element[OListe4, k] + Element[UListe5, k]], k, 1, 6)
UListe9 = Sequence(Simplify[(Element[Liste7, k] * Element[UListe6, k])], k, 1, 6)
k = 2 (slider from 0 to 5, step 1)
p = 0.19 (slider from -5 to 5, step 0.01)
a = 0 (slider from 0 to 12, step 0.1)
b = 7.6000000000000005 (slider from 0 to 12, step 0.1)
c: x = a
d: x = b
f(x) = (p * x^(k))
l: b - a
Δx: b / n
Liste1 = Sequence(a + (Δx * j), j, 0, n)
Liste2 = Sequence((a + (Δx * j), 0), j, 0, n)
U_Rechtecke = Sequence(Polygon[((j * Δx), 0), (((j + 1) * Δx), 0), (((j + 1) * Δx), f((j * Δx))), ((j * Δx), f((j * Δx)))], j, 0, n - 1)
O_Rechtecke = Sequence(Polygon[(((j - 1) * Δx), 0), ((j * Δx), 0), ((j * Δx), f((j * Δx))), (((j - 1) * Δx), f((j * Δx)))], j, 1, n)
Text1: "" + (LaTeX[LaTeX[Δx, false, true]]) + "=" + (LaTeX[LaTeX[Δx, true, false]]) + ""
Text2 = LaTeX(O_Rechtecke, false, true)
Text4 = LaTeX(U_Rechtecke, false, true)
U_S = LowerSum(f, a, b, n)
O_S = UpperSum(f, a, b, n)
Text5: "" + (LaTeX[LaTeX[O_S, false, true]]) + "=" + (LaTeX[LaTeX[O_S, true, false]]) + ""
Text6: "" + (LaTeX[LaTeX[U_S, false, true]]) + "=" + (LaTeX[LaTeX[U_S, true, false]]) + ""
Liste3 = Sequence(f(a + (j * Δx)), j, 0, n)
s1: ((b - a) * a^(3)) + (((3 * a^(2)) * (b - a)^(2) / n^(2)) * Sum[Sequence[j, j, 0, n - 1]])
s2: (((3 * a) * (b - a)^(3) / n^(3)) * Sum[Sequence[j^(2), j, 0, n - 1]])
s3: ((b - a)^(4) / n^(4) * Sum[Sequence[j^(3), j, 0, n - 1]])
s_kO = Element(OListe4, k + 1)
s_01: s_0
ff(n, k) = 1 - n^(k)
s_kU1 = Element(UListe9, k + 1)
h = Simplify(Element[OListe4, 1] + Element[UListe5, 1])
OListe8 = Sequence(Simplify[(Element[Liste7, k] * Element[OListe4, k])], k, 1, 6)
s_kO1 = Element(OListe8, k + 1)
OListe10 = Sequence(Limit[Element[OListe8, k], Infinity], k, 1, 6)
OListe11 = Sequence(FractionText[Element[OListe10, k]], k, 1, 6)
UListe10 = Sequence(Limit[Element[UListe9, k], Infinity], k, 1, 6)
UListe11 = Sequence(FractionText[Element[UListe10, k]], k, 1, 6)
g(k) = ((p * 1 / (k + 1)) * b^(k + 1))
Kontrollwert = Integral(f, 0, b)
c_1 = 0
U(n) = ((p * b^(k + 1)) * s_kU1(n))
U_n: U(n)
O(n) = ((p * b^(k + 1)) * s_kO1(n))
O_n: O(n)
Text8: "\text {" + (LaTeX[n]) + " Intervalle der Länge Δx=" + (LaTeX[Δx]) + " zwischen " + (LaTeX[n + 1]) + " Punkten}"
Text11: "\int_{0}^{b}{f(x)dx }=\int_{0}^{" + (LaTeX[b]) + "}{f(x)dx }= " + (LaTeX[LaTeX[Kontrollwert, false, false]]) + "=" + (LaTeX[Kontrollwert]) + ""
s_kU = Element(UListe6, k + 1)
allgemeinFormel = false
Text10: "\int_{0}^{b}{ f(x)dx }=\int_{0}^{" + (LaTeX[b]) + " }{ f(x)dx }=  \lim_{ n \to \infty }U(n) =    \lim_{ n \to \infty } p  \cdot b^" + (LaTeX[k + 1]) + "  \cdot \left(  " + (LaTeX[s_kU1]) + "  \right )= p  \cdot b^ " + (LaTeX[k + 1]) + " \cdot   \lim_{ n \to \infty } \left(  " + (LaTeX[s_kO1]) + " \right )=     p  \cdot b^" + (LaTeX[k + 1]) + "    \cdot \frac{1}{" + (LaTeX[k + 1]) + "  }    =  p   \cdot \frac{1}{" + (LaTeX[k + 1]) + "  }   \cdot b^ " + (LaTeX[k + 1]) + "     =   " + (LaTeX[p]) + "         \cdot \frac{1}{ " + (LaTeX[k + 1]) + "  }   \cdot  " + (LaTeX[b]) + "^  " + (LaTeX[k + 1]) + "    = " + (LaTeX[g(k)]) + "                 "
Text9: "\int_{0}^{b}{ f(x)dx }=\int_{0}^{" + (LaTeX[b]) + "}{ f(x)dx }= \lim_{ n \to \infty }O(n) =     \lim_{ n \to \infty }   p  \cdot b^" + (LaTeX[k + 1]) + "  \cdot \left(  " + (LaTeX[s_kO1]) + "    \right)    =    p  \cdot b^    " + (LaTeX[k + 1]) + "  \cdot   \lim_{ n \to \infty } \left(  " + (LaTeX[s_kO1]) + "  \right)     =  p  \cdot b^  " + (LaTeX[k + 1]) + " \cdot \frac{1}{" + (LaTeX[k + 1]) + " }     =  p   \cdot \frac{1}{" + (LaTeX[k + 1]) + " }   \cdot b^ " + (LaTeX[k + 1]) + "   =        " + (LaTeX[p]) + "   \cdot \frac{1}{ " + (LaTeX[k + 1]) + " }   \cdot" + (LaTeX[b]) + "^ " + (LaTeX[k + 1]) + "    =   " + (LaTeX[g(k)]) + "    "
O_1 = Sum(O_Rechtecke)
U_S2 = Sum(U_Rechtecke)
O_S2 = Sum(O_Rechtecke)
Text20_1: "k = " + (LaTeX[k]) + ""
Position2 = (3, 7.2)
Text7: "O(n)= \sum_{j=1}^{ n}   {Δx \cdot f(j\cdot Δx )= \sum_{j=1}^{n}{Δx \cdot p \cdot (j\cdot Δx )^ " + (LaTeX[k]) + "=p  \cdot  Δx^" + (LaTeX[k + 1]) + "   \cdot\sum_{j= 1}^{n }{ j^" + (LaTeX[k]) + " }=p  \cdot  \frac{b^" + (LaTeX[k + 1]) + " }{n^" + (LaTeX[k + 1]) + "}\cdot   \sum_{j=1}^{n}{ j^ " + (LaTeX[k]) + " } = p  \cdot  \frac{b^" + (LaTeX[k + 1]) + "  }{n^" + (LaTeX[k + 1]) + "  }\cdot  \left(  " + (LaTeX[s_kO]) + " \right)  }= p  \cdot b^" + (LaTeX[k + 1]) + "  \cdot \left(  " + (LaTeX[s_kO1]) + "     \right) }=" + (LaTeX[O_n]) + ""
Text3: "U(n)= \sum_{j=0}^{n-1}{Δx \cdot f(j\cdot Δx )= \sum_{j=0}^{n-1}{Δx \cdot p \cdot (j\cdot Δx )^" + (LaTeX[k]) + "=p  \cdot  Δx^ " + (LaTeX[k + 1]) + "\cdot  \sum_{j=0}^{n-1}{ j^" + (LaTeX[k]) + "}=p  \cdot  \frac{b^" + (LaTeX[k + 1]) + "}{n^" + (LaTeX[k + 1]) + "}\cdot   \sum_{j=0}^{n-1}{ j^" + (LaTeX[k]) + " }= p  \cdot  \frac{b^" + (LaTeX[k + 1]) + " }{n^ " + (LaTeX[k + 1]) + "}\cdot  \left( " + (LaTeX[s_kU]) + "    \right) } = p  \cdot b^" + (LaTeX[k + 1]) + " \cdot \left( " + (LaTeX[s_kU1]) + "    \right) }    =" + (LaTeX[U_n]) + "         "
Text13: "f(x)=\textcolor{#800000} p\cdot x^\textcolor{#009900}k =\textcolor{#800000}{" + (LaTeX[p]) + "}\cdot x^\textcolor{#009900}" + (LaTeX[k]) + ""