Integration

The integral originated in attempts to find areas of various shapes and the ideas involved in
finding integrals are much older than the ideas related to finding derivatives. In fact,
Archimedes^{1}
was finding areas of various curved shapes about 250 B.C. using the main ideas of the
integral. What is presented here is a generalization of these ideas. The main interest is in the
Riemann integral but it is easy to generalize to the so called Riemann Stieltjes integral in
which the length of an interval,

is replaced with an expression of the form
g

−g

where
g is a function of finite total variation, a special case being the Riemann integral when
g

=
x. In this chapter are the principal theorems about Stieltjes integrals including the
Riemann integral as a special case. A good source for more of these things this is the book by
Apostol, [2].

In all which follows we will always tacitly assume that f is a bounded function defined on an appropriate finite interval.

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