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Math Help - dense ball

  1. #1
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    dense ball

    A solid spherical ball of radius R is created in a way such that the density at the point (x,y,z) is proportional to the point's distance from the origin.
    What is the mass of the ball?
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  2. #2
    fgn
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    I hope you didn't see my first solution. This time I have actually done some thinking.
    The density  \delta(r) = kr for some  0 < k

    The surface of a sphere with radius  r is  r  = 4\pi r^2
    A sphercial shell  dV = 4\pi r^2\,dr
     dm = \delta\,dV
     dm =  4k\pi r^3\,dr
    Thus,
    m = 4k\pi\int_0^R r^3\,dr = \pi kR^4
    Last edited by fgn; December 8th 2006 at 03:36 AM.
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  3. #3
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    Quote Originally Posted by putnam120 View Post
    A solid spherical ball of radius R is created in a way such that the density at the point (x,y,z) is proportional to the point's distance from the origin.
    What is the mass of the ball?
    The rule is,
    m=\int \int_S \int \rho(x,y,z) dV
    In this case,
    \rho(x,y,z)=\kappa \sqrt{x^2+y^2+z^2}=\kappa \rho

    Use spherical coordinates,
    \int_0^{2\pi} \int_0^{\pi} \int_0^R \kappa \rho^3 \sin \phi d\rho\, d\phi\, d\theta
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