## Riemann Integral

The Riemann integral is the Integral normally encountered in Calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue Integral), but are unlikely to be encountered outside the confines of advanced mathematics texts.

The Riemann integral is based on the Jordan Measure, and defined by taking a limit of a Riemann Sum,

 (1)

 (2)

 (3)

where and , , and are arbitrary points in the intervals , , and , respectively. The value is called the Mesh Size of a partition of the interval into subintervals .

As an example of the application of the Riemann integral definition, find the Area under the curve from 0 to . Divide into segments, so , then

 (4) (5) (6)

By induction
 (7)

so
 (8)

 (9)

For example, take .
 (10)
so
 (11)

Kestelman, H. Riemann Integration.'' Ch. 2 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 33-66, 1960.