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Math Help - Triple Integral Problem Using Spherical Coordinates

  1. #1
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    Triple Integral Problem Using Spherical Coordinates

    Given that:

    Integral[-a, 0] Integral[-Sqrt(a^2-y^2), 0] Integral[a-Sqrt(a^2-x^2-y^2), a] dz dx dy

    http://tinypic.com/view.php?pic=4ibucl&s=6

    Please write an equivalent integral in spherical coordinates.
    Last edited by dynas7y; December 3rd 2009 at 03:25 PM.
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  2. #2
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    bump
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  3. #3
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    Hi Dynas, Looks to me your integral:

    \int_{-a}^0 \int_{-\sqrt{a^2-y^2}}^0 \int_{a-\sqrt{a^2-x^2-y^2}}^a dz dx dy

    is the integral from the bottom of the sphere surface in the plot below, to the plane z=a shown in blue. Now, just because the integrand f(x,y,z)=1, that's the same as just the volume of 1/8 of a sphere of radius a in spherical coordinates:

    \int_{0}^{\pi/2}\int_0^{\pi/2} \int_0^a \rho^2 \sin(\phi) d\rho d\theta d\phi=(1/8) 4/3 \pi a^3
    Attached Thumbnails Attached Thumbnails Triple Integral Problem Using Spherical Coordinates-sphere-section.jpg  
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  4. #4
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    Thanks a lot. That was a big help.
    Last edited by dynas7y; December 4th 2009 at 02:11 PM.
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