Comparison of riemann and lebesgue integral

Abstract

The development of the integral is most introductory analysis course is centered almost exclusively on the Riemann integral. In this historical development the integration is simply introduced as finding the area under a curve. The Riemann integration is a basic concept in mathematical analysis, since it related to boundedness, continuity and differentiability. We also consider some integrals of Stieltjes types which are considered as generalization of the Riemann Integrals which involves two bounded functions. The Stiltjes integral has very useful applications in probability theory, mechanics as well as theoretical physics. Another theory of integration more general than the Riemann theory was called Lebesgue integral, it consider the concept of measure of a set, starting with simple function and ending with measurable function, this approach leads to greater generality in the types of function that can integrated. We will compare both of this integration by using their theorem.