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Thread: Finding Mass

  1. #1
    Junior Member
    Nov 2007

    Finding Mass

    I am having trouble with the following question:

    Find the mass of a ball B given by (x^2) + (y^2) + (z^2) <= (a^2) if the density at any point is proportional to its distance from the z-axis.

    Any ideas or help would be appreciated. Thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
    Feb 2009
    You need to change variables. Rectangular is painful. The density here is
    $\displaystyle kr=k\sqrt{x^2+y^2}=k\rho\sin\phi$, but the k's cancel, so you can let k=1.
    Try spherical, where the Jacobian is $\displaystyle \rho^2\sin\phi$.
    If you're integating over the entire ball, then $\displaystyle 0<\rho <a$, $\displaystyle 0<\theta <2\pi$ and $\displaystyle 0<\phi<\pi$.
    Last edited by matheagle; Mar 14th 2009 at 05:42 PM.
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