Problem

Find the volume of the solid generated by rotating the region R bounded by the y axis, the line y = a, and the curve

around the x axis.

Solution

Problems like this are always easier to solve using pictures. Here is the region R.

If R is rotated around the x axis, the generated solid is a solid cylinder with a parabolic hole scooped in the middle. A horizontal cross section looks like a washer with inside radius

and outside radius a.

The height (thickness) of the cross section is Dx, and the area is

So the total volume of the solid is the sum of infinitely many thin cross sections of area p(a² – x), giving us