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Mathematics-Online lexicon:

Riemann Integral

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The definite integral of a piecewise continuous function $ f$ 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 $ [a,b]$;

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


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


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  automatically generated 9/22/2016