Win a Free Year of College from Sophia.
Enter Now!

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.

Our College Algebra Course is only $329.

Sophia's online courses not only save you money, but credits are also eligible for transfer to over 2,000 colleges and universities.*

*The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 16 of Sophia's online courses. More than 2,000 colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

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