The integral is an old idea-a venerable part of the cakulus .The integralis important because（i）it is dual to the derivative（as the Fundamental Theorem of Cakulus tells us）and（ii）it is the most powerful and versatile tool in calculus. Most of real analysis is based on the theory of the integral. And the integral， in turn， inspires many of the key ideas of functional analysis.
　　There are many theories of the integral. One of the first modern theories was that enunciated by Bernhard Riemann in his celebrated paper on Fourier series. The Riemann integral is relatively simple， and it does the job nicely It is certainly the most ubiquitous and widely-used integral in the world today. Mathematicians， physicists， engineers， and many others use the Riemann integral regularly.