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Surface Integral

  1. Feb 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the volume integral of the function [tex]f=x^{2}+y^{2}+z^{2}[/tex] over the region inside a sphere of radius R, centered on the origin.

    2. Relevant equations

    Spherical polars [tex]x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta)[/tex]

    Jacobian in spherical polars = [tex]r^2sin(\theta)[/tex]

    3. The attempt at a solution

    When i work through it I end up with the triple integral

    [tex]V=\int^{R}_{0}dr\int^{\pi}_{-\pi}d\phi\int^{\pi}_{-\pi}d\theta (r^{2}sin^{2}\theta cos^{2}\phi+r^{2}sin^{2}\theta sin^{2}\phi + r^{2}cos^{2}\theta)r^2sin\theta[/tex]

    but i'm not too sure whether this is right. Mainly i'm not sure about the limits of integration.

    Is this correct please?

  2. jcsd
  3. Feb 19, 2009 #2


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    If you want to integrate over all theta, that's only theta in [-pi/2,pi/2], isn't it? And you can simplify the integrand a LOT. x^2+y^2+z^2=r^2.
  4. Feb 19, 2009 #3
    Ah yeah I didn't bother simplifying the integrand but I can see I should have done cos it would have made it a lot easier to type lol. I thought about it some more and understand why the limits are as you said now. Thanks!
  5. Feb 19, 2009 #4


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    Is there are reason why you titled this "surface integral"?
  6. Feb 19, 2009 #5
    Because I was tired :D Oops lol
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