The Shell Method - Calculus 2

So far we have seen how to use definite integrals to calculate the volume of solids of revolution with the disk and washer method. However, they are not the only methods that can be used to calculate that volume. There is an alternative method. Instead of using thin disks or thin washers to represent the volume of these solids, we can use cylindrical shells via the shell method.

The shell method calculates the volume of solids of revolution by adding up the volume of an infinite number of thin cylindrical shells (a cylinder with a hole) that would make up the solid. This can be done for solids formed by revolving a region around the x-axis, y-axis, or any other horizontal or vertical line.

What makes the shell method useful in comparison to the disk/washer method is that when using the shell method, you work in terms of the opposite variable that you would with the disk/washer method. Revolving around a horizontal axis? The shell method will work in terms of y while the disk/washer method works in terms of x. Revolving around a vertical axis? The shell method will work in terms of x while the disk/washer method works in terms of y. Because of this, there are certain situations where the shell method is more preferable to use over the disk/washer method and vice versa.

There is a lot to know regarding the shell method, so let's get started!

In this lesson, you will learn:

• How to use the shell method to calculate volume of solids of revolution formed by revolving a region around the x-axis and y-axis
• How to use the shell method to calculate the volume of solids of revolution formed by revolving a region around a line other than the x-axis or y-axis
• The advantages to the shell method in comparison to the disk/washer method

I hope you find this video to be helpful!

-Josh