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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