Mathematik-Online lexicon: Example of a Riemann Integral

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	Mathematik-Online lexicon:
	Example of a Riemann Integral

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To compute $\int_0^1 x^2\,dx$ with Riemann sums, we choose a sequence of partitions

$\displaystyle \Delta_n: x_i= i/n,\quad i=0,\ldots,n \,,$

with points

$\displaystyle \xi_i=(2i-1)/(2n),\quad i=1,\ldots,n \,.$

The Riemann sums are

$\displaystyle \int f_{\Delta_n}$	$\displaystyle =$	$\displaystyle \sum_{i=1}^n \frac{1}{n}\left(\frac{2i-1}{2n}\right)^2 = \frac{1}{4n^3} \left(4\sum_{i=1}^n i^2-4 \sum_{i=1}^n i + \sum_{i=1}^n 1\right)$
	$\displaystyle =$	$\displaystyle \frac{1}{4n^3} \left( \frac{4n(n+1)(2n+1)}{6} - \frac{4n(n+1)}{2} +n \right) = \frac{1}{3} -\frac{1}{12n^2} \,,$

and

$\displaystyle \lim_{n\to\infty} \int f_{\Delta_n} = \frac{1}{3} \,,$

in accordance with the exact value of the integral.

see also:

automatisch erstellt am 22. 9. 2016