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Math Help - Not Able To Obtain The Same Answer In Spherical/Cylindrical Coordinate

  1. #1
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    Not Able To Obtain The Same Answer In Spherical/Cylindrical Coordinate

    Hi,

    I have to solve this problem using spherical coordinate and validate my answer using cylindrical coordinate, but I have 2 different answers, obviously!

    Can anyone tell me what I am doing wrong? (See attached file for the problem and what I did)

    Thanks, Mike.

    EDIT: sorry, I used the wrong angle for the limit of \varphi, it should be \frac{\pi}{6} not \frac{\pi}{3}.

    My new answer is 0,420894

    EDIT 2: I drew the solid in my cad software and came up with a volume of 0,561.

    I tried to do it the long way, calculating the volume of the cone, sphere and cylinder.

    H of cone (z) = 4-\sqrt{3(0^2+2^2)}=4-.535898=3.4641

    Cone: V= (\frac{1}{3}\pi*2^2*3.4641)=14.5104
    Sphere: V= (\frac{4}{3}\pi*2^3)=\frac{16\pi}{3}
    Cylinder: V= (\pi*2^2*.535898)=6.7343

    Total volume = Cone + Cylinder - Sphere = 14.5104 -  \frac{16\pi}{3} + 6.7343 = 4,48953
    Since I only need from \frac{\pi}{4} to \frac{\pi}{2} (one eight) and not 2 \pi, I divide by 8.

     \frac{4,48953}{8}=0,561191
    Attached Files Attached Files
    Last edited by MoneyHypeMike; July 3rd 2011 at 07:57 PM. Reason: Adding information
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  2. #2
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    Re: Not Able To Obtain The Same Answer In Spherical/Cylindrical Coordinate

    For the spherical coordinates case, your limits for \rho look wrong. Firstly, if you're above the sphere, the lower limit should be 2. Secondly, you have not converted the Cartesian equation for the cone correctly into sphericals. The equation \phi =\pi/3 is a cone with vertex at the origin; this is not what you were given.
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