How to Find the Volume of a Solid between Two Surfaces of Revolution

If you want to find the volume of a solid that falls between two different surfaces of revolution, you can use the meat-slicer method to do this. The meat-slicer method works best with solids that have similar cross sections. Here’s the plan:

1. Find an expression that represents the area of a random cross section of the solid in terms of x.

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

3. Evaluate this integral.

The trick is to find a way to describe the donut-shaped area of a cross section as the difference between two integrals: one integral that describes the whole shape minus another that describes the hole.

For example, suppose that you want to find the volume of the solid shown here.

A vase-shaped solid between two surfaces of revolution.

This solid looks something like a bowl turned on its side. The outer edge is the solid of revolution around the x-axis for the function

The inner edge is the solid of revolution around the x-axis for the function

Here’s how to solve this problem:

1. Find an expression that represents the area of a random cross section of the solid.

   That is, find the area of a circle with a radius of

   

   and subtract the area of a circle with a radius of

   

2. Use this expression to build a definite integral that represents the volume of the solid.

   The limits of integration are 0 and 4:

   

3. Solve the integral:

   

   Now evaluate this expression:

   

Here’s another problem: Find the volume of the solid shown here.

Another solid formed between two surfaces of revolution.

This solid falls between the surface of revolution y = ln x and the surface of revolution

bounded below by y = 0 and above by y = 1.

The cross section of this solid is shown on the right-hand side of the figure: a circle with a hole in the middle.

Notice, however, that this cross section is perpendicular to the y-axis. To use the meat-slicer method, the cross section must be perpendicular to the x-axis. Modify the problem using inverses:

The resulting problem is shown in this figure.

Use inverses to rotate the problem from the earlier figure so you can use the meat-slicer method.

Now you can use the meat-slicer method to solve the problem:

1. Find an expression that represents the area of a random cross section of the solid.

   That is, find the area of a circle with a radius of e^x and subtract the area of a circle with a radius of

   

   This is just geometry. Remember that the area of a circle is πr²:

   

2. Use this expression to build a definite integral that represents the volume of the solid.

   The limits of integration are 0 and 1:

   

3. Evaluate the integral:

   

   So the volume of this solid is approximately 9.179 cubic units.