We use the procedure of “Slice, Approximate, Integrate” to develop the shell method to compute volumes of solids of revolution.

The shell method

Some volumes of revolution require more than one integral using the washer method. We study such an example now.

The shell method formula

Let’s generalize the ideas in the above example. First, note that we slice the region of revolution parallel to the axis of revolution, and we approximate each slice by a rectangle. We call the slice obtained this way a shell. Shells are characterized as hollow cylinders with an infinitesimal difference between the outer and inner radii and a finite height. We now summarize the results of the above argument.

Now that we have established the above result, we do not have to go through the “Slice, Approximate, Integrate” procedure for every example. Let’s see now how the formula works in action.