Integral Calculus, An Introduction
As mentioned in the
introduction to calculus,
integral calculus began long before differential calculus,
although the latter is presented first in most text books.
In the third century B.C.,
Aristotle became interested in the area bounded by certain curves.
He used rectangles to approximate these regions,
and then used smaller and smaller rectangles,
so that the approximation became better and better.
He didn't have a formal theory of limits,
but he usually new where the numbers were heading.
Thus he computed the area under many common curves.
He called this procedure the
"method of exhaustion".
generalized this procedure,
and made it rigorous,
using the concepts of limits that were developed for differential calculus.
The "Riemann sum", described next,
is similar to Aristotle's rectangles,
but the rectangles need not have a uniform thickness.
Also, Riemann's method generalizes to higher dimensions,
e.g. computing the volume under a surface.
Lebesgue introduced a more general approach to integration,
that is based on measure theory.
However, it is easier to master Riemann integration first,
then move on to Lebesgue integration;
so let's get started.