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## Riemann 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 | overview |

The definite integral of a piecewise continuous function is defined by

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 :

i.e.,

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

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automatisch erstellt am 22. 9. 2016 |