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Math Help - Volume Problem - "Shell" Method

  1. #1
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    Volume Problem - "Shell" Method

    Hi again friends,

    I would appreciate help on 2 more problems please.

    1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating about the y-axis and the region R bounded by the curve y=2x^2-x^3 (two ex squared minus ex cubed) and the x-axis.

    2. Use the method of cylindrical shells to find the volume of the solid obtained rotating the region R by the curves y = x^3 (ex cubed), y = 0 and x = 2 and about the line x = 3
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, nmq3b!

    Can you sketch the region, or do you have a graphing calculator?
    A good sketch is absolutely essential.


    1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating
    about the y-axis, the region R bounded by the curve y\:=\:2x^2-x^3 and the x-axis.
    Code:
                |
           *    |        *
                |     *:::::*
            *   |    *:::::::*
             *  |  *::::::::::
          - - - * - - - - - - * - -
                |             2
                |              *
                |               *
    Shells Formula: . V \;=\;2\pi\int^b_a\text{(radius)(height)}\,dx

    In this problem: . \text{radius} \:=\:x,\;\text{height} \:=\:2x^2-x^3

    Hence: . V \;=\;2\pi\int^2_0x(2x^2-x^3)\,dx

    I assume you can finish it . . .



    2. Use the method of cylindrical shells to find the volume of the solid obtained rotating
    the region R boounded by the curves y\:=\:x^3,\;y = 0,\;x = 2 about the line x = 3
    Code:
                |       *   :
                |       |   :
                |      *|   :
                |     *:|   :
                |  .*:::|   :
        - - - - * - - - + - + - -
            *   |       2   3
          *     |
         *      |

    This time we have: . \text{radius } = 3-x,\;\text{height} = x^3

    Hence: . V \;=\;2\pi\int^2_0(3-x)x^3\,dx

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