Theory

Theory 1

Take a graph in the first quadrant of the -plane. Rotate this about the -axis. The resulting 3D body is symmetric around the axis. We can find the volume of this body by using an integral to add up the volumes of infinitesimal shells, where each shell is a thin cylinder.

The volume of each cylindrical shell is :

In the limit as and the number of shells becomes infinite, their total volume is given by an integral.

Volume by shells - general formula

In any concrete volume calculation, we simply interpret each factor, ‘’ and ‘’ and ‘’, and determine and in terms of the variable of integration that is set for .

Shells vs. washers

Can you see why shells are sometimes easier to use than washers?