One of the standard integration problems is to find the volume of a
solid of revolution. For a curve y=f(x) that runs from x=a to
x=b, if we rotate around the axis y=AR, the lines from the curve to the
axis of rotation becomes a disc. The cross sections have area
DiskArea(x) = pi*(f(x)-AR)2. The volume of the region is .
If we rotate the other way, around the y axis, the line from the
top curve to the bottom curve becomes a shell of height (f(x)-g(x) and
radius x. The volume of the region is .