Question

Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise.
The circulation line integral of F=<(2xy^2),(4x^3)+y> where C is the boundary of {(x,y): 0<=y<=sinx, 0<=x<=pi}

Answer 1

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

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