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# Riemann Integral

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The definite integral of a piecewise continuous function is defined by Here is a partition of ; denotes the maximal length of the interval and is an arbitrary point in the -th interval. The sums on the side of the integral's definition are called Riemann sums. For a positive function , corresponds to the area below the graph of .

Slightly strengthening the hypotheses, we assume that is continuously differentiable, and prove the convergence of the Riemann sums using Cauchy's criterion.

For a sequence of partitions with , we compare the Riemann sums corresponding to and with the aid of a partition , which consists of the union of the points of and : The difference of the Riemann sum for , to the Riemann sum for can be estimated via the mean value theorem. From we obtain       i.e., With an analogous estimate for , Cauchy's criterion follows: for The convergence of two sequences to the same limit can be established with an identical argument.

For piecewise continuous , the proof is technically more complicated. It relies on the uniform continuity of on the closed interval of integration: für [Back]

 automatisch erstellt am 22.  9. 2016