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Math Help - Help with Triple Integration in Cylindrical Coordinates

  1. #1
    Junior Member
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    Sep 2009
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    Help with Triple Integration in Cylindrical Coordinates

    I'm having trouble with the following problems:

    1.) Find the volume of the region bounded by the paraboloids z = 2x^2 +y^2 and z= 12 -x^2 -2y^2

    2.) Find the volume of the region bounded above by the spherical surface x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 +y^2

    Can someone give an explanation of what the boundaires would be for triple integration in cylindrical coordiantes
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  2. #2
    Senior Member
    Joined
    Dec 2008
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    To find the boundaries of integration, we find where the two surfaces intersect. For instance,

    \begin{aligned}<br />
2x^2+y^2&=12-x^2-2y^2\;\;\;\;\;\mbox{when}\\<br />
3x^2+3y^2&=12\\<br />
x^2+y^2&=4,<br />
\end{aligned}

    which means that our two paraboloids intersect in a circle of radius 2. Our integral is therefore

    \int_0^{2\pi}\int_0^2\int_{2r^2\cos^2\theta+r^2\si  n^2\theta}^{12-r^2\cos^2\theta-2r^2\sin^2\theta}r\,dz\,dr\,d\theta.
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