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Green's Theorem to evaluate the line integral

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Use Green's Theorem to evaluate the line integral of the vector field F along the given positively oriented curve C.

    F(x,y) = <sin(x^3) +x^2(y), 3xy-(x)(y^2)+e^(y^2)> and C is the boundary of the region enclosed by the semicircle y = √(4-x^2) and the x-axis.

    2. Relevant equations



    3. The attempt at a solution

    I did the problem, but I just need someone to check my work.

    Its hard to explain what I did in this forum, but I did the partial derivative of x from 3xy-(x)(y^2)+e^(y^2) and the partial of y from sin(x^3) +x^2(y). I subtracted these two and got 3y-y^2 - x^2.

    Then I converted to polar coordinates. The domain is from 0<θ<pi and 0<r<2 and the function becomes 3(r^2)cosθ-r^3. I evaluated and got -8pi.
     
  2. jcsd
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