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We have the following $$ \int_{\Gamma}-x^2ydx+xy^2dy$$ Where $\Gamma$ is the semi circle given by $x^2+y^2=a^2$, $y>0$. We are to demonstrate Green's theorem. My attempt is as follow: Write $$\int_{\Gamma}\left(\begin{array}{c}-x^2y\\xy^2\\\end{array}\right).\left(\begin{array}{c}dx\\dy\\\end{array}\right)$$ And parameterising the semi circle by $ x= a\cos{t}$ and $ y = a\sin{t}$ for $0\leq t\leq \pi$ We get: $$ 2a^4\int_{0}^\pi\cos^2{t}\sin^2{t}dt = \frac{a^4\pi}{4}$$ And now applying Green's theorem we get: $$\int_{-a}^{a}\int_{0}^{\sqrt{a^2-x^2}}(x+y)dydx=\int_0^{\pi}\int_{0}^{a}r^2(\cos{\theta}+\sin{\theta})drd\theta = \frac{2a^3}{3}$$

I feel I'm missing something obvious. Many thanks in advance.

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1 Answer 1

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The Green's theorem should give you

$$\iint_D(\frac{\partial (xy^2)}{\partial x}-\frac{\partial (-x^2 y)}{\partial y}) dxdy=\iint_D (x^2+y^2) dxdy=\int^{\pi}_0 \int^a_0 r^3 dr d\theta$$ which should give you the same result as the other one.

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  • $\begingroup$ I see where I have gone wrong! Many thanks for your help. $\endgroup$
    – user211337
    Commented Feb 25, 2015 at 21:00

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