Online Test Banks
Score higher
See Online Test Banks
Learning anything is easy
Browse Online Courses
Mobile Apps
Learning on the go
Explore Mobile Apps
Dummies Store
Shop for books and more
Start Shopping

How to Evaluate the Volume of a Solid of Revolution

You can evaluate the volume of a solid of revolution. A solid of revolution is created by taking a function, or part of a function, and spinning it around an axis — in most cases, either the x-axis or the y-axis.

A solid of revolution of <i>y</i> = 2 sin <i>x</i> around the <i>x</i>-axis.
A solid of revolution of y = 2 sin x around the x-axis.

For example, the left side of the figure shows the function y = 2 sin x between x = 0 and


Every solid of revolution has circular cross sections perpendicular to the axis of revolution. When the axis of revolution is the x-axis (or any other line that’s parallel with the x-axis), you can use the meat-slicer method directly.

However, when the axis of revolution is the y-axis (or any other line that’s parallel with the y-axis), you need to modify the problem.

To find the volume of this solid of revolution, use the meat-slicer method:

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

    This cross section is a circle with a radius of 2 sin x:

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

    This time, the limits of integration are from 0 to π/2:

  3. Evaluate this integral by using the half-angle formula for sines:


    Now evaluate:


So the volume of this solid of revolution is approximately 9.8696 cubic units.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus

Inside Sweepstakes

Win $500. Easy.