The Riemann integral is a method of integration which approximates the integral as lower and upper limits of a sum of terms. For the lower limit we split the region of integration up into a sequence of intervals, take the lowest value ofon each interval, multiply this value by the length of the interval, and add the resulting terms. For the upper limit we take the largest value ofon each interval, multiply this value by the length of the interval, and add the resulting terms. If the functionis Riemann integrable, the the upper and lower limits are equal as the number of intervals tends to infinity and the length of each interval tends to zero. We must make this rigorous.

Letbe an interval ofand letbe a bounded function. For any finite set of pointssuch thatthere is a corresponding partition of

Letbe the set of all partitions ofwithThen letbe the infimum of the set of upper Riemann sums with each partition inand letbe the supremum of the set of lower Riemann sums with each partition inIfthen sois decreasing andis increasing. Moreover,and are bounded byTherefore, the limitsandexist and are finite. Ifthenis Riemann-integrable overand the Riemann integral ofoveris defined by

Example: Show thatis Riemann integrable over

is an increasing function. If we split the region of integrationinto n equal intervals each of lengththen each interval isexcept for the upper end interval which is

In the intervaland in the interval therefore

HenceLetthen the upper and lower limits are equal -- sois Riemann integrable overand