Next, we need to find where the two curves intersect, which will be the endpoints of the
region. We do this by setting the two curves equal to each other. We get 3x^2 − 4 = x^3 . The
solutions are (−1, −1) and (2, 8). Therefore, in order to find the area of the region, we need to
evaluate the integral (x^3 − (3x^2 − 4)) dx = (x^3 − 3x^2 + 4) dx. We get (x^3 − 3x^2 + 4) dx
= = − = .
2.
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^3 − y^2 and g(y) = 2y.
First, let’s make a sketch of the region.
