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Homework Help: Derive the volume of a sphere.

  1. Oct 19, 2005 #1
    The forumula for 2¶r can intergrated to make ¶r^2 (at least I think). So can anyone derive the volume of a sphere
    4/3¶r^3?
     
  2. jcsd
  3. Oct 19, 2005 #2

    dextercioby

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    First thing: the volume of a sphere is...0.

    Second: to find the volume of a ball seen as a domain in [itex] \mathbb{R}^{3} [/itex], one could antidifferentiate the expression giving the surface of the ball as a function of its radius.

    Daniel.
     
  4. Oct 19, 2005 #3

    HallsofIvy

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  5. Oct 19, 2005 #4
    to derive the volume of a sphere I think you want to start with the unit ball in 3 space. Then convert to spherical coordinates and integrate the triple intergral. I think im close, I remember doing this a while back so take it FWIW.
     
  6. Oct 20, 2005 #5
    Why not try a surface of revolution? For example, take a semi-circle and rotate it around the x-axis to find the volume.

    [tex]\text{V}=\pi\int_{a}^{b}f(x)^2\,dx[/tex]
     
  7. Oct 21, 2005 #6
    I solved this out for practice on my own. I found the volume of a revolved surface. I said [tex]\text{V}=\pi\int_{r}^{-r} (\sqrt{r^2 - x^2})^2dx[/tex].
    I'm pretty sure that should work thus giving you a simple to evaluate integral of
    [tex]\pi\int_{r}^{-r} \(r^2 -x^2)dx[/tex].
    Should be easy enough. If you have trouble with understanding whereabout the integral came from, realize that it is simply the sum of the volume's of cylinders.
    (Note: I hope I typed that latex code right...I've never used it before. Sorry if it comes out wrong.)
     
    Last edited: Oct 21, 2005
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