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Math Help - volume of a solid

  1. #1
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    volume of a solid

    The region bounded by  y = e^{-x^{2}}, y = 0, x = 1, and is revolved about the y-axis. Find the volume of the resulting solid.

    im not quite sure how to do this one
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by viet View Post
    The region bounded by  y = e^{-x^{2}}, y = 0, x = 1, and is revolved about the y-axis. Find the volume of the resulting solid.

    im not quite sure how to do this one
    are you sure those are the right limits? as it is, the area is not bounded. maybe it should be x=0 and x = 1 and the x-axis?
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  3. #3
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    I think you want unbounded area.
    Consinder a long interval [0,N].

    Then the volume is given by,
    2\pi \int_0^N xe^{-x^2} dx

    To find the unbounded area (which is actually bounded).

    Find,
    2\pi \int_0^{\infty} xe^{-x^2} dx

    Hint: Use t=-x^2.
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  4. #4
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    sorry i seem to forgot something, the correct question is:

    The region bounded by  y = e^{-x^{2}}, y = 0, x = 0, x = 1 and is revolved about the y-axis. Find the volume of the resulting solid.
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by viet View Post
    sorry i seem to forgot something, the correct question is:

    The region bounded by  y = e^{-x^{2}}, y = 0, x = 0, x = 1 and is revolved about the y-axis. Find the volume of the resulting solid.
    that's what i thought you meant to say, thought it probably would be cool to do it TPH's way if you didn't make a typo

    We proceed by the Method of Cylindrical Shells:

    V = 2 \pi \int_{0}^{1} xe^{-x^2} dx

    Let u = -x^2

    \Rightarrow du = -2x dx

    \Rightarrow - \frac {1}{2} du = x dx

    So our integral becomes:

    V = - 2 \pi \cdot \frac {1}{2} \int_{x=0}^{x=1} e^u du

    \Rightarrow V = - \pi \left[ e^u \right]_{x=0}^{x=1} = - \pi \left[ e^{-x^2} \right]_{0}^{1}

    \Rightarrow V = \pi - \frac { \pi}{e}
    Attached Thumbnails Attached Thumbnails volume of a solid-vol2.jpg  
    Last edited by Jhevon; May 24th 2007 at 07:47 PM.
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