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Surface integrals - parametrizing a part of a sphere

  1. Apr 9, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the area of the part of the sphere x^2 + y^2 + z^2 = 4z
    that lies inside the paraboloid x^2 + y^2 = z

    2. Relevant equations

    3. The attempt at a solution
    I solved for the intercepts and found that they are z=0 and z=3.
    The sphere is centered two units in the z-direction above the origin.

    Hence I wanted to switch to spherical coordinates and got:


    Now, since we know that the sphere is centered on (0,0,2) we can take find the angle from the Z axis easily.


    The Surface Area of this portion of the sphere should then become
    [itex]\int^{2∏}_{0} \int^{∏}_{∏/3} 4SinΦ dΦdΘ[/itex]
    \int^{2∏}_{0} \int_{∏/3}_{∏} 4SinΦ dΦdΘ

    I get 12∏, which is incorrect. The correct answer is 4∏. Where am I going wrong?
    Last edited: Apr 9, 2014
  2. jcsd
  3. Apr 9, 2014 #2
    That itex thing never works for me either :/
  4. Apr 9, 2014 #3
    Nevermind I solved it. I was taking the wrong portion of the curve.
  5. Apr 9, 2014 #4


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    Gold Member

    In the tex, don't use special characters; instead use the tex code for them. Quote this to see how I changed your code.$$
    \int_0^{2\pi}\int_{\pi/3}^{\pi} 4\sin\theta d\phi d\theta$$
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