You can solve many challenging volume problems using the disk/washer method. There are, however, some volume problems that are difficult to solve with this method. For this reason, we extend our discussion of volume problems to the shell method, which—like the disk/washer method—is used to compute the volume of solids of revolution.
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Key: The representative element (rectangle) must be parallel to the axis of revolution. Recall that for discs or washers, the element had to be perpendicular to the axis of revolution.
Remember: PARASHELL vs PERPENDISCULAR Revolving rectangular elements about a parallel axis produces cylindrical shells (like the wrappings around a toilet paper roll). In the equation to the right, x is the radius of shell and f(x) - g(x) is the height of the shell. |
Question: Why would using the disk method for this problem be much harder?
Answer: Two integrals are needed, plus you would need to solve the equation for x.
Answer: Two integrals are needed, plus you would need to solve the equation for x.
Example 3
Use both the shell method and and the disk method to find the volume formed by revolving the region bounded by y = x^3, x = 2, and y =0 about the x-axis.
Use both the shell method and and the disk method to find the volume formed by revolving the region bounded by y = x^3, x = 2, and y =0 about the x-axis.
Note: Both methods give the same answer, but the disk method is much easier. If the method is not specified, choose the one the is easiest to do.
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This website was prepared by Brant Breeding in his personal capacity. These materials are not endorsed, approved, sponsored, or provided by or on behalf of the University of Arkansas, Fayetteville.
Images on this page courtesy of Briggs, Gillet, and Schulz and Taylor (full citation found on Reference page).
Please follow this link for our full disclaimer statement.