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Math Help - ellipsoid intersected by cylinder

  1. #1
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    ellipsoid intersected by cylinder

    Find volume of the ellipsoid x^2 +2(y^2+z^2) \le 10 intersected by the cylinder y^2 = z^2 \le 1

    seems like changing to cylindrical coordinates would be best

    so I have

    \left\{ \begin{array}{cc} r^2cos\theta + 2r^2sin^2\theta +2z^2 \le 10 \\r^2sin^2\theta +z^2 \le 1 \end{array}\right.

    0 \le \theta \le 2\pi

    and solving for z -\sqrt{1-r^2sin^2\theta} \le z \le \sqrt{1-r^2sin^2\theta}

    and -\sqrt{8}sec\theta \le r \le \sqrt{8}sec\theta

    before I set up the integrals, are those the correct bounds?

    edit: I tried on wolfram|alpha but it seems to not work
    Last edited by Jonroberts74; July 25th 2014 at 12:47 PM.
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  2. #2
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    Re: ellipsoid intersected by cylinder

    Treating the x-axis as the z-axis


    x^2 +2r^2(cos^2\theta + sin^2\theta)=10

    r^2(cos^2\theta + sin^2\theta)=1

    0 \le \theta \le 2\pi ; -1 \le r \le 1 ; -\sqrt{10-2r^2} \le x \le \sqrt{10-2r^2}

    \int_{0}^{2\pi}\Bigg(\int_{-1}^{1}\Bigg(\int_{-\sqrt{10-2r^2}}^{\sqrt{10-2r^2}}dx\Bigg)rdr\Bigg)d\theta

    but I get an answer of 0 when I check it so something is still amiss
    Last edited by Jonroberts74; July 26th 2014 at 12:17 PM.
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  3. #3
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    Re: ellipsoid intersected by cylinder

    r bounds were wrong I got it
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