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Using the methods you have learned, find the volume of the region bounded by: , , and , revolved about the lines below. Use the steps outlined in problem 8-49.

  1. -axis

    After sketching an arbitrary rectangle (with ), it is apparent that there will not NOT be a hole in the center of this solid.
    That indicates that the Disk Method will be appropriate.

  2. -axis

    This arbitrary rectangle indicates that there will be a hole in the center of this solid. You will have to use the Washer Method.

    Horizontal rectangles indicate that the bounds and the integrand will be in terms of .

    Notice that the figure will be a cylinder with a curvy hole in the center.
    The radii of the cylinder is .
    The radii of the curvy hole is .

  3. Q: Washers or disks? Think: Will there be a hole or won't there?

    Q: Will the integral be in terms of or ? Think: Are the rectangles vertical or horizontal?

    Notice that these radii are units longer than the radii in part (b).