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Solid of Revolution - Shell Method

Solid of Revolution - Shell Method


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.

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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.

The Shell Method

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

Source: self-created video

Examples of applying the shell method

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

Problem Set

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 y equals 1 half x squared, 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 y equals x squared 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 straight y equals sin left parenthesis x right parenthesis space a n d space y equals e to the power of x squared end exponent minus 1. 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