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Homework Help: Surface Integral using Divergence Theorem

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Evaluate the surface Integral [tex]I=\int\int_S\vec{F}\cdot\vec{n}\,dS[/tex]

    where [tex]\vec{F}=<z^2+xy^2,x^100e^x, y+x^2z>[/tex]

    and S is the surface bounded by the paraboloid [itex]z=x^2+y^2[/itex]

    and the plane z=1; oriented by the outward normal.





    3. The attempt at a solution

    [tex]I=int\int_S\vec{F}\cdot\vec{n}\,dS=\int\int\int_E(div\vec{F})dV[/tex]

    [tex](div\vec{F})=y^2+x^2[/tex]

    [tex]\Rightarrow I=\int\int_D(\int_{z=x^2+y^2}^1(x^2+y^2)\,dz)\,dA[/tex]

    [tex]\Rightarrow I=\int\int_D(1-(x^2+y^2)\,dA[/tex]

    Is it just Polar Coordinates all the way home now?

    Thanks,
    Casey
     
  2. jcsd
  3. Dec 7, 2008 #2

    Dick

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    No, I don't think that's right. The integral of f(x,y)dz is z*f(x,y), isn't it? Try evaluating that between the limits again.
     
  4. Dec 7, 2008 #3
    I am sorry, where are you referring to?
    The divergence of F= x^2+y^2 That is f(x,y), now if I integrate that wrt z
    ohh! I forgot to include the factor of f(x,y) so it should be

    [tex] I=\int\int_D(x^2+y^2)*(1-(x^2+y^2)\,dA[/tex]

    yes? Now Polar?
     
  5. Dec 7, 2008 #4

    Dick

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    Now polar.
     
  6. Dec 7, 2008 #5
    Polar bears. Polar beers. Sweet numchuck skills Dick. Thanks.
     
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