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 as

Thus, we break the integral into intervals and , and in both cases we are taking the integral of , but the definition of will depend on which interval we are in. The computation is as follows:

1) on line 1, the second integral’s integrand is x – 1 – x^2 + 2x, instead of x – 1 – x^2 – 2x.

2) on line 2, the first integral’s integrand should be -x^2 + x + 1, instead of -x^2 – x + 1

you could have translated the graph 1unit to the left , and it would be the same as problem 11