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lys-0071 [83]
3 years ago
15

Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise.

Mathematics
1 answer:
anzhelika [568]3 years ago
3 0

By Green's theorem, the circulation of \vec F(x,y)=\langle2xy^2,4x^3+y\rangle around C is equal to

\displaystyle\iint_D\frac{\partial(4x^3+y)}{\partial x}-\frac{\partial(2xy^2)}{\partial y}\,\mathrm dx\,\mathrm dy

where D is the region with C as its boundary. The integral is equivalent to

\displaystyle\int_0^\pi\int_0^{\sin x}(12x^2-4xy)\,\mathrm dy\,\mathrm dx=\boxed{\frac{23\pi^2}2-48}

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