The Shell method is a method of computing the volume of a surface of revolution. It is also commonly referred to as the method of cylindrical shells. This method is most convenient when, for instance, a region in the x,y-plane below the graph of a function of x is being revolved about the y-axis, or a region below the graph of a function of x is being revolved about the y-axis. In such cases, it is usually preferable to the most familiar method, the disk method.

To derive the shell-method formula, first suppose that we have a nonnegative continuous function f(x) defined on an interval [a,b]. (The derivation is analogous for a non-positive function, or for a function of y instead of x.) Suppose that we are revolving the area below the graph of f (and above the x-axis) around the y-axis, and...