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Green's Theorem for a circle

  1. Sep 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Use Green's Theorem to calculate second type line integral:
    [tex]I = \oint_{\Gamma} x^2 y dx - xy^2dy[/tex]
    where [itex]\Gamma[/tex] is the edge of domain [itex] D = \left\{(x,y) | x^2 + y^2 \leq 1, y \geq 0 \right\}[/itex]

    2. Relevant equations

    Green's Theorem.

    3. The attempt at a solution

    Ok, so according to Green's Therom this integral becomes:
    [tex]-\iint_{D} x^2 + y^2 dA[/tex]
    And since my domain is the upper part of a radius 1 circle centered at the origin, we can make it like this:
    [tex]-\int_{0}^{\pi} \int_{0}^{1}(\cos^2{\theta} + \sin^2{\theta}) r dr d\theta = -\int_{0}^{\pi} \int_{0}^{1} r dr d\theta = -\frac{\pi}{2}[/tex]

    But for some reason I got a feeling that something went wrong here, maybe because of the minus sign in front of [itex]\frac{\pi}{2}[/itex]?
    Was I correct in making the transformation to polar coordinates after constructing the double integral, or should it had been better before (even though the differentiation becomes much more complicated then)?
     
  2. jcsd
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