Let

Find the area between the graphs of and on the interval .

First, we draw the graph, shading the region between the two graphs in blue.

In this problem on the entire interval ; however, the function is defined piecewise on this interval.

The functions and are defined piecewise on the interval by

and

So, we break each integral into two pieces and compute:

Truly I believe that it is better to just use basic geometry for this problem. I think Apostol intended you to, as well, considering he earlier spoke of “the double-edged sword” of using an integral to calculate area, and sometimes using geometry to calculate an integral.