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Help doing an integral using stokes theorem?

  1. Apr 19, 2012 #1
    1. The problem statement, all variables and given/known data
    F= xi + x3y2j + zk
    C is the boundary of the semi-ellipsoid z=√(4-4x2-y2) in the plane z=0


    2. Relevant equations

    Stokes theorem states:
    ∫∫(curlF ° n)dS

    3. The attempt at a solution
    I found the curl of the F to be 3x2y2k
    I found that the dot product of CurlF and n = 3x2y2 divided by dS

    Then,
    ∫∫(3x2y2)/(dS)*dS
    =∫∫(3x2y2)dxdy
    I evaluated this integral on Wolfram Alpha with the following boundaries:
    y goes from -√(4-4x2) to √(4-4x2)
    and x from -1 to 1 and got the correct answer = ∏

    However, I am finding it impossible (for me!) to do the integral by hand and am wondering if someone can help me turn this into polar coordinates or something else that makes it more solvable

    Thank!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 20, 2012 #2
    You want to integrate on the region [itex]x^2 + \left( \frac{y}{2} \right)^2 \leq 1[/itex]. Consider the substitution [itex]y = 2y_1[/itex]. Then your integral becomes:

    [tex]\int_{D} 24 x^2 y_1^2\ dx dy_1[/tex]

    Where D is now the unit disk in the (x, y_1) plane centered at the origin. Can you see how to put that into polar coordinates?
     
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