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Math Help - Shell method Volume of a solid problem

  1. #1
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    Shell method Volume of a solid problem

    Here's the problem:

    Use the Shell Method to compute the volume of the solids obtained by rotating the region enclosed by the graphs of the functions , and about the y-axis.

    I can solve problems when I am given one or two functions, an interval, and an axis to rotate about...but I am completely lost on how to solve this kind of problem. Any help would be great, thanks>
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  2. #2
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    Hello, calnix!

    Did you make a sketch?


    Use the Shell Method to compute the volume of the solid obtained
    by rotating the region bounded by: y \:=\:x^2,\;y \:=\:8-x^2,\;x \:=\: \tfrac{3}{2} about the y-axis.
    Code:
                      |
          *           *     |     *
                 *    |    *|
           *  *       |     |:*  *
            *         |     |:::*
           * *        |     |::*:*
               *      |     |*  :
          *       *   |   * |   : *
        - - - - - - - * - - + - + - -
                      |    1   2
                      |

    Formula: . V \;=\;2\pi \int^b_a x\left(y_2 - y_1\right)\,dx

    The two parabolas intersect at x = \pm2


    We have: . V \;=\;2\pi\int^2_{\frac{3}{2}} x\left[(8-x^2) - x^2\right]\,dx

    Go for it!

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  3. #3
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    Nov 2008
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    Thanks a lot! I didn't know I'd use the given x-value and then find the other for the bounds. Appreciate it a lot! If you don't mind, I have another problem:

    Use both the Shell and Disk Methods to calculate the volume of the solid obtained by rotating the region under the graph of for about:
    the x-axis:____
    the y_axis:____

    I need to find what values of the axis the region is rotated around. Would I just use the 0 and 13/3 for the bounds?
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