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Multiple Integral


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

We compute the integral of the function

$\displaystyle f(x,y)=xy
$

over the domain

$\displaystyle V:\, 0\le x\le 1,\qquad 0\le y\le 1+x^2
$

as limit of Riemann sums (approximation with step functions).

\includegraphics[width=0.6\linewidth]{bsp_integral1}

Using a square grid with grid-width $ h=1/n$ leads to the Riemann sum

$\displaystyle h^2 \sum_{0\le j<n}\ \sum_{0\le kh <1+(jh)^2} (jh)(kh)
=
h^4 \sum_{0\le j<n} j \sum_{0\le k <n+j^2/n} k
\,.
$

Neglecting terms of higher order, and noting that

$\displaystyle \sum_{0\le\ell<r} \ell^m = r^{m+1}/(m+1) + O(r^m)
\,,
$

yields
    $\displaystyle \frac{1}{n^4}\,\sum_{0\le j<n} j\left((n+j^2/n)^2/2+O(n)\right)$  
    $\displaystyle \qquad
=\frac{1}{n^4}\,\sum_{0\le j<n} jn^2/2 + j^3 + j^5/(2n^2)+O(n^2)$  
    $\displaystyle \qquad
=
\left( \frac{1}{4} + \frac{1}{4} + \frac{1}{12}\right)
+ O(\underbrace{1/n}_{h})
\,.$  

Passing to the limit $ n\to\infty$, we obtain $ 7/12$ as value of the integral.

see also:


  automatisch erstellt am 22.  9. 2016