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Math Help - Find mass and center of mass of the solid S....

  1. #1
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    Find mass and center of mass of the solid S....

    bounded by the paraboloid z=4x^2+4y^2 and the plane z=a (a>0) if S has constant density K.

    I just need someone to explain to me how to set up the integral using cylindral coordinates.
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  2. #2
    Senior Member
    Joined
    Dec 2008
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    We first note that x_M=y_M=0. To find z_M using cylindrical coordinates, we let

    \begin{aligned}<br />
x&=r\cos\theta\\<br />
y&=r\sin\theta\\<br />
z&=z.<br />
\end{aligned}

    Our paraboloid equation becomes

    z=4r^2,

    and the equation of intersection of the paraboloid with the plane z=a becomes

    \begin{aligned}<br />
4r^2&=a\\<br />
r&=\frac{\sqrt{a}}{2}.<br />
\end{aligned}

    Our integral becomes

    z_M=\frac{1}{M}\int_0^{2\pi}\int_0^{\frac{\sqrt{a}  }{2}}\int_0^a zKr\,dz\,dr\,d\theta,

    where

    M=\int_0^{2\pi}\int_0^{\frac{\sqrt{a}}{2}}\int_0^a  Kr\,dz\,dr\,d\theta.
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