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.
