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Math Help - integral (within sphere, below cone)

  1. #1
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    integral (within sphere, below cone)

    Find the volume of the solid that lies within the sphere , above the plane, and outside the cone .

    I am having trouble finding the integral of phi

    p is from 0 to 1
    theta is from 0 to 2pi
    but the scale of 3 on the function is screwing me up with theta

    i feel like im getting close, but i need a push
    i thought it may have been 3*sqrt(1/10) to pi/2
    but it didnt turn out to be
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  2. #2
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    Well, we just want the volume of the cone with the sphere top then just subtract it out of the sphere right? Equation of the cone is z=3\sqrt{x^2+y^2}. So in spherical coordinates that's \rho \cos(\phi)=3\sqrt{\rho^2\sin^2(\phi)}. Do all that and get: \rho^2\left(9\sin^2(\phi)-\cos^2(\phi)\right)=0 or \phi=\arctan(1/3). So equation of the cone in spherical coordinates is \rho=\arctan(1/3)=a. Do same with the sphere to get equation of it in spherical coordinates \rho=1. Anyway, turn the crank and get:

    V_{\text{cone}}=\int_0^a\int_0^{2\pi}\int_0^1 \rho^2\sin(\phi)d\rho d\theta d\phi

    Pretty sure anyway. Double check it ok.
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  3. #3
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    im going to have a very busy day today, but ill make sure to check it out tomorrow for sure
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