The integral originated in attempts to find areas of various shapes and the ideas involved in finding integrals are much older than the ideas related to finding derivatives. In fact, Archimedes1 was finding areas of various curved shapes about 250 B.C. using the main ideas of the integral. What is presented here is a generalization of these ideas. The main interest is in the Riemann integral but it is easy to generalize to the so called Riemann Stieltjes integral in which the length of an interval,
In all which follows we will always tacitly assume that f is a bounded function defined on an appropriate finite interval.