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Proof of Volume of a ball

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data

    http://staff.washington.edu/dhlee528/003.JPG [Broken]

    2. Relevant equations

    x = r sin ( phi) cos ( theta)

    y = r sin ( phi )sin (theta)

    z = r cos ( phi )


    3. The attempt at a solution

    [tex]
    vol=8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}\int_0^r \rho^2 \sin(\phi)d\rho d\theta d\phi
    [/tex]

    [tex]
    8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2} \sin(\phi)(\frac{\rho^3}{3}){|}_0^r d\theta d\phi
    [/tex]

    [tex]
    \frac{4r^3 \pi}{3}\int_0^\frac{\pi}{2}sin(\phi)d\phi
    [/tex]

    [tex]
    -\frac{4r^3\pi}{3}[0-1]=\frac{4\pi r^3}{3}
    [/tex]



    I think I got spherical coordinate right but don't know how to do for rectangular or spherical coordinate
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Mar 3, 2010 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi dhlee528! Welcome to PF! :smile:

    (have a theta: θ and a phi: φ and a pi: π :wink:)

    For rectangular coordinates: obviouly the volume element is dxdydz, so decide which order you're going to integrate in … say keep z and y fixed, decide the limits on x; then keep z fixed, decide the limits on y.

    What do you get? :smile:
     
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