For the readers of “A Case of Pre-FTC Definite Integral“, this post illustrates yet another way of evaluating the following definite integral without FTC:
Let us divide into unequal subintervals (see Fig. 1) such that
in general and
so that the subintervals become finer as increases.
We have, for the sum of the rectangles above the curve ,
Since as . Consequently,
It follows that