Mathematics-Online lexicon: Riemann Integral

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	Mathematics-Online lexicon:
	Riemann Integral

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The definite integral of a piecewise continuous function

is defined by

$\displaystyle \int_a^b f(x)\,dx = \lim_{\vert\Delta\vert\to0} \int_a^b f_\Delta = \lim_{\vert\Delta\vert\to0} \sum_{k} f(\xi_k)\,\Delta x_k \quad .$

Here $\Delta:\,a=x_0<x_1<\cdots<x_n=b$ is a partition of

;

$\displaystyle \vert\Delta\vert=\max_k \Delta x_k\,, \qquad \Delta x_k=x_k-x_{k-1}\,,$

denotes the maximal length of the interval and $\xi_k$ is an arbitrary point in the

-th interval. The sums on the side of the integral's definition are called Riemann sums.

$\includegraphics[width=0.6\linewidth]{riemann_bild}$

For a positive function , $\int_a^b f$ corresponds to the area below the graph of .

Example:

automatically generated 9/22/2016