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Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C cos(y) dx + x2 sin(y) dy C is the rectangle with vertices (0, 0), (5, 0), (5, 4), (0, 4)

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By Green's theorem,


\displaystyle\int_C\cos y\,\mathrm dx+x^2\sin y\,\mathrm dy=\iint_D\left((\partial(x^2\sin y))/(\partial x)-(\partial(\cos y))/(\partial y)\right)\,\mathrm dx\,\mathrm dy

where
D is the region with boundary
C, so we have


\displaystyle\iint_D(2x+1)\sin y\,\mathrm dx\,\mathrm dy=\int_0^5\int_0^4(2x+1)\sin y\,\mathrm dy\,\mathrm dx=\boxed{60\sin^22}

User Chauncy Zhou
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