Volume: The Shell Method
Cylindrical Shell Method
If the cross sections of the solid are taken parallel to the axis of Revolution, then the cylindrical shell method will be used to find the volume of the solid. If the cylindrical shell has radius r and height h, then its volume would be 2 πrh times its thickness. Think of the first part of this product, (2 πrh), as the area of the rectangle formed by cutting the shell perpendicular to its radius and laying it out flat. If the axis of revolution is vertical, then the radius and height should be expressed in terms of x. If, however, the axis of revolution is horizontal, then the radius and height should be expressed in terms of y.
Learn more about the shell method by reading the explanation below.
Practice: Find the volume using the shell method in the following exercises.