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Homework Help: Triple Integration of a Sphere in Cylindrical Coordinates

  1. Apr 2, 2009 #1
    1. The problem statement, all variables and given/known data

    The problem was to find the volume enclosed by a sphere of radius "a" centered on the origin by crafting a triple integral and solving for it using cylindrical coordinates.

    2. Relevant equations

    [tex]x^{2}+y^{2}+z^{2}=a^{2}[/tex] : Equation for a sphere of radius "a" centered on the origin.

    [tex]\iiint\limits_E dV[/tex] : Triple integral for finding volume of a region [tex]E[/tex].

    3. The attempt at a solution

    I solved the triple integral (but I don't think it's right) and got this: [tex]\frac{4}{3}a^{2}\pi[/tex] ---> Actually, I think I solved the integral right, but I think my bounds are incorrect.

    I used the following as my bounds and subsequent iterated integral:

    [tex]E=\{ \ (r,\theta,z) \ | \ 0\leq r\leq a, \ 0\leq \theta\leq 2\pi, \ -\sqrt{a^2-r^2}\leq z\leq \sqrt{a^2-r^2} \ \}[/tex]

    [tex]\int^{2\pi}_{0}\int^{a}_{0}\int^{\sqrt{a^2-r^2}}_{\sqrt{a^2-r^2}} dz dr d\theta[/tex]

    If my proposed answer isn't right could the problem lie within my bounds? I'm not really great at determining the bounds for iterated integrals yet >.<'

    Thanks :D
  2. jcsd
  3. Apr 2, 2009 #2


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    Hi Zarlucicil! :smile:

    (have a pi: π and a theta: θ and try using the X2 tag just above the Reply box :wink:)

    erm :redface: … it isn't dz dr dθ, it's … ? :smile:
  4. Apr 2, 2009 #3
    Well, your bounds are in fact correct. What's missing is a volume element, namely, you need to have [tex]r drd\theta dz[/tex]. But maybe you just missed it, since without it you probably won't get the final answer you mentioned (which isn't correct by the way - you need [tex]4\pi a^3/3[/tex]).

    You can also write [tex]r[/tex] in terms of [tex]z[/tex], which is in my opinion a bit more intuitive. In that case you get:

    [tex]\int^{2\pi}_{0}\int^{-a}_{a}\int^{0}_{\sqrt{a^2-z^2}} r dr dz d\theta[/tex]

    Gives the same answer ofcourse.
  5. Apr 4, 2009 #4
    Ahhh I see. Thanks for the replies, I understand what I did wrong now :D. I can't believe I missed the volume element, ughhh. O well.
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