Notice that in the region from x = −1 to x = 0 the top curve is f(x) = 0 (the x-axis), and the
bottom curve is g(x) = x^3 , but from x = 0 to x = 2 the situation is reversed, so the top curve is
f(x) = x^3 , and the bottom curve is g(x) = 0. Thus, we split the region into two pieces and find
the area by evaluating two integrals and adding the answers: (0 − x^3 ) dx and (x^3 − 0) dx.
We get (0-x^3 ) dx = = 0 − = and (x^3 − 0) dx = −
0 = 4. Therefore, the area of the region is .
5.
We find the area of a region bounded by f(y) on the right and g(y) on the left at all points of the
interval [c, d] using the formula [f(y) − g(y)] dy. Here, f(y) = y + 2 and g(y) = y^2.
First, let’s make a sketch of the region.
