Question

find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=x^2+1; y=9-x^2; about y=-1

Answer 1

First find the intersections between the two curves, they are the limits of integration. They can be found readily as x=±2.

Next, we check that the curves do not cross the axis y=-1 between the limits of integration. Otherwise the limits of integration must change.

Now use the washer method to calculate the area, since the "outer" radius R is based on the upper curve, and the inner radius r is on the lower curve.
The integration formula is then
Area = π∫(R(x)^2-r(x)^2)dx