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Homework Help: Volume of a sphere derivation

  1. Oct 1, 2005 #1
    I just need a really good derivation of it using spherical coordinates, like the integral limits.

    pictures might help
     
  2. jcsd
  3. Oct 1, 2005 #2
    [tex]\iiint\limits_E{\rho}^2\,\sin{\phi}\,d\phi\,d\rho\,d\theta\quad E:\left\{0\leq\phi\leq\pi;\quad 0\leq\rho\leq r;\quad 0\leq\theta\leq 2\pi\right\}[/tex]
     
    Last edited: Oct 1, 2005
  4. Oct 1, 2005 #3
  5. Oct 1, 2005 #4
    actually i was more interested in how you derived the d phi(that other angle thing) part

    Like which integrant belongs to which. Mathworld doesnt show too much of that, the math part I get but I would like to know which angle belong to which. Since there are 3 sets of integral limits, then there should 3 of them, so which belongs which accoring to the equation cavoy posted
     
  6. Oct 1, 2005 #5
    From cartesian to spherical coordinates:

    [tex]x=\rho\cos{\phi}\cos{\theta}[/tex]

    [tex]y=\rho\cos{\phi}\sin{\theta}[/tex]

    [tex]z=\rho\sin{\phi}[/tex]

    ...then use the Jacobian to get the equivalent of dV in terms of phi, theta, and rho.
     
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