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Math Help - volume of revolving solid

  1. #1
    jsl
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    volume of revolving solid

    Hi! I am having some trouble with this one. Any help you can give would be great! Thanks!

    Find the volume of the solid generated by revolving about the -axis the region bounded by the upper half of the ellipse
    and the -axis, and thus find the volume of a prolate spheroid. Here and are positive constants, with .
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    Quote Originally Posted by jsl View Post
    Hi! I am having some trouble with this one. Any help you can give would be great! Thanks!




    Find the volume of the solid generated by revolving about the -axis the region bounded by the upper half of the ellipse



    and the -axis, and thus find the volume of a prolate spheroid. Here and are positive constants, with .
    You know what the graph of the ellipse looks like right? When you rotate this about the x-axis, you create a continuum of circlular cross-sections from x=a to x=-a. The area of these cross-sections are given by \pi y^2=A(x)=\pi\left(b^2-\frac{b^2x^2}{a^2}\right).

    Just plut A(x) into:
    V=\int_{-a}^aA(x)dx=2\int_0^aA(x)dx.

    V=2\pi b^2\int_0^a\left(1-\frac{x^2}{a^2}\right)dx

    Does this make sense?
    Last edited by adkinsjr; December 3rd 2009 at 03:15 PM.
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  3. #3
    jsl
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    Thank you! That was exactly what I needed. And I discovered that I had been most of the way there, so your help was just what I needed to finish it up!
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