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The diagram at right shows the region bounded by the -axis, , , and . The region is revolved about the -axis to create the solid shown with dotted lines. .

  1. Describe a method you can use to determine the volume of the solid.

    Will you use Washers or Disks? Will the bounds and integrand be written in terms of or ?

  2. Set up and evaluate the integrals needed to calculate the volume. (Using washers, the solution will require two integrals.)

    The outside of this solid will be a cylinder with radius .

    The inside of the solid will have a hole. But what shape is that hole? Is it cylindrical?

    Notice that the bottom of the hole is cylindrical (with radius ), while the top of the hole is determined by

    Consequently, you will need to use the Washer Method twice... both of which are rotated about the -axis.

2 separate Solid curves, left starting @ (negative 1, comma 0.5) & passing through (negative 3, comma 4.5), right starting at (1, comma 0.5) & passing through (3, comma 4.5), & dashed cylinder, diameter of bottom base on the x axis, from x = negative 3 to x = 3, diameter of top base at, y = 4.5, also from x = negative 3 to x = 3, with dashed inner cylinder, diameter of bottom base on the x axis from x = negative 1 to x = 1, diameter of top base @ y = 0.5, from x = negative 1 to x = 1.

Use the eTool below to help solve the problem.
Click on the link to the right to view the full version of the eTool. Calc 8-63 HW eTool