Author:
Richard Enderton

Students should be able to calculate the volume of a solid of revolution, applying the shell method.

This packet consists of an instructional video, a video with examples, a problem set, and other resources.

Tutorial

The student should be familiar with the concept of using the definite integral as a summation, and should also be familiar with the disk and washer methods for finding the volume of a solid of revolution.

This video develops the concept and integral formula for calculating the volume of a solid of revolution using cylindrical shells.

Source: self-created video

This video walks through several examples of applying the shell method and contains instruction on how to set up the shell method for various situations.

Source: self-created video

1. Use both disk (or washer) AND shell method to find the volume of the solid of revolution if the region R is revolved about the given axis.

R is bounded by , the x-axis and x = 4.

a. Revolve R about the y-axis. (201.062)

b. Revolve R about the line x = 7. (268.083)

c. Revolve R about the line y = 10. (509.357)

d. Revolve R about the x-axis. (160.85)

2. The first quadrant region bounded by and y=4 is revolved about the y=axis.

a. Find the volume of the resulting solid. (25.133)

b. A hole is drilled in the solid. The hole is centered along the y-axis. What must the radius of the hole be to remove one-quarter of the volume of the solid? (0.732)

3. R is bounded by . Find the volume of the solid generated when R is revolved about the line x = -2. (1.517)

4. A torus (donut) has a cross section with radius 1. The center of the cross section is 4 units from the center of the torus. Use the concept of solid of revolution and the shell method to calculate the volume of this torus. (78.957)

Source: own problems

Paul’s Online Notes

http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx

Master Math Mentor

http://www.mastermathmentor.com/calc/abcalc.ashx

Shell Method Demo Gallery

http://www.mathdemos.org/mathdemos/shellmethod/gallery/gallery.html

Wolfram Alpha