On Review of the Riemanns’ Iterated Integrals

This paper reviews the Riemanns’ approach of multiple integrals by considering its basic definitions, results and the various successive integral evaluations, in doing this, the continuity condition of R was considered with respect to X as the necessary condition guaranteeing the existence of the integrals using the analytical Riemanns’ approach. This involves the use of partitions, to generate continued results to the nth iterated integral after the nth iteration.

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On the Review of Riemann and Riemann Stieljtes Integrals

This work seeks an understanding of integration as a generalization of the summation process either in the Riemann or the Riemann Stieljtes sense as the case may be.

First, on having an interval [a, b] in ℝ, a partition is constructed with which the Riemann sums R(f, p) is calculated and if such sums tends to a finite  limit and the mesh m(p) tends to zero then the function is intergrable and the Riemann integration of such function is defined as

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

where U(f, p) and L(f, p)  have their usual meanings.

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

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

Further, we explored the various properties of the Riemann and Riemann Stieljtes integrals in form of theorems and lemma and in the last section we stated and proved some important and advanced results on the subject.

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