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.
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

image1.png

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

image2.png

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

    image3.png

    and subtract the area of a circle with a radius of

    image4.png
  2. Use this expression to build a definite integral that represents the volume of the solid.

    The limits of integration are 0 and 4:

    image5.png
  3. Solve the integral:

    image6.png

    Now evaluate this expression:

    image7.png

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

Another solid formed between two surfaces of revolution.
Another solid formed between two surfaces of revolution.

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

image9.png

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:

image10.png

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.
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 ex and subtract the area of a circle with a radius of

    image12.png

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

    image13.png
  2. Use this expression to build a definite integral that represents the volume of the solid.

    The limits of integration are 0 and 1:

    image14.png
  3. Evaluate the integral:

    image15.png

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

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