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Math Help - a double integral question

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    Question a double integral question

    The sphere of radius a centered at the origin is expressed in rectangular coordinates as x^2 + y^2 +z^2 = a^2 , and hence its equation in cylindrical coordinates is r^2 + z^2 = a^2 . Use this equation and a polar integral to find the volume of the sphere.

    Thank you very much.
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  2. #2
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    Quote Originally Posted by kittycat View Post
    The sphere of radius a centered at the origin is expressed in rectangular coordinates as x^2 + y^2 +z^2 = a^2 , and hence its equation in cylindrical coordinates is r^2 + z^2 = a^2 . Use this equation and a polar integral to find the volume of the sphere.

    Thank you very much.
     \iiint_V \ dV = 2\int_0^{2\pi} \int_0^a \int_{0}^{\sqrt{a^2-r^2}} r \  dz \ dr \ d\theta
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  3. #3
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    hi perfecthacker,
    Thank you very much for your reply. I haven't learnt triple integral. Could you please explain this question to me in double integral. Thanks.
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  4. #4
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    \int_{0}^{2\pi}\int_{-a}^{a}r\cdot{r}drd{\theta}

    \int_{0}^{2\pi}\int_{-a}^{a}r^{2}drd{\theta}
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  5. #5
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     V = 2 \int_{0}^{2 \pi} \int_{0}^{a} \sqrt{a^2-r^2} r \ dr \ d \theta

     V = 2 \int_{0}^{2 \pi} \left[\frac{1}{3}(a^2-r^2)^{3/2} \right] from  0 to  a .

     V = \frac{2}{3} \int_{0}^{2 \pi} a^3 \ d \theta

     V = \frac{4 \pi}{3} a^3
    Last edited by tukeywilliams; August 1st 2007 at 03:01 PM.
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