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Surface integral without using Gauss' theorem

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Calculate §§ A.n dS if
    A= 2y(x^2)i-(y^2)j + 4xzk
    over the region in the first octant bounded by (y^2)+(z^2) = 9 and x = 2

    2. Relevant equations



    3. The attempt at a solution

    Let n = (yj + zk) / 3

    then A.n = [-(y^3) +4xz^3] / 3

    Since we 'll project the surface onto the xy-plane:
    |n.k| = z/3 and z = SQRT(9-y^2)

    Putting all together I obtain
    = §§R (4xz^3 - (y^3))/z dx dy



    Now making the appropriate changes and setting up the limits of integration:


    §y=30 §x=20 4x(9-y^2) - (y^3)/sqrt(9-y^2) dx dy



    However I always obtain 108 as a result and not 180 as my book suggested me (and after verification by Gauss' divergence theorem.

    Is there a problem with the limits of integration? Wrong projection? I really have no clue ...
    Thanks for the help!
     
  2. jcsd
  3. Sep 20, 2009 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    You've only calculated the integral over one side/face of the surface....there are three more faces that make up the closed surface bounding the given region...you need to calculate the surface integral over all 3 of those as well.
     
  4. Sep 20, 2009 #3
    Thanks a lot!
    I finally got it (at least I hope so ;-) !
     
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