Mathematik-Online lexicon: Multiple Integral

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	Mathematik-Online lexicon:
	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

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 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

as value of the integral.

see also:

automatisch erstellt am 22. 9. 2016