On the Review of Riemann and Riemann Stieljtes Integrals | Asian Research Journal of Current Science

This research aims to comprehend integration as a generalisation of the summation process, either in the Riemann or Riemann Stieljtes sense, depending on the case.

First, given an interval [a, b] in R, a partition is created with which the Riemann sums R(f, p) are calculated, and if such sums tend to a finite limit and the mesh m(p) goes to zero, the function is intergrable, and its Riemann integration is defined as

A refinement of p for any partition p R and p' in [a, b] R and p'

where U(f, p) and L(f, p) have the same meaning as before.

Again, using the interval [a, b] and a partition on it, compute the Riemann Stieljtes sums of f with respect to a, R (f, p, a), and if the sums tend to a finite limit as the mesh m(p) to zero, the function is Riemann Stieljtes integrable, and such integral is defined as

for all p and p' partitions in [a, b], and p' a refinement of p, where U(f, p', a) and L(f, p, a) have their usual meanings

We next used theorems and lemma to investigate the many properties of the Riemann and Riemann Stieljtes integrals, and in the final section, we stated and proved some important and advanced results on the subject.

Please see the link :- https://globalpresshub.com/index.php/ARJOCS/article/view/794