## 4.8 Volumes of Solids of Revolution

### Summary

Integrals find application in many modeling situations involving continuous variables such as area. They allow us to model physical entities that can be described through a process of adding up, or accumulating, smaller infinitesimal parts. In this section, the Riemann Sum approach is used to develop an integral formula for the volume of a solid of revolution.

By the end of your studying, you should know:

• How to develop the formula for the volume of a solid of revolution.
• The integral formula for the volume of a solid of revolution.
• How to use the integral formula to compute the volume of a solid of revolution.

On-screen applet instructions: The applet depicts approximating the volume of a solid of revolution with a finite number n of disks. Use the pull-down menu to change the value of n.

### Examples

Find the volume of the solid generated by rotating the region R bounded by the y axis, the line y = a, and the curve

Find the volume of the solid generated by rotating the region bounded by y = x, y = 3 – x, and x = 4 around the line x = 5.

Find the volume of the torus of radius a with inside radius b.

Volume By Disks

Volume By Shells

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 4.8 Volumes of Solids of Revolution (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Solids of Revolution
Tripod

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel