[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]

Mathematics-Online lexicon: Annotation to

# 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

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