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

The line integral you need to compute is

$\displaystyle\int_C\langle2xy^2,4x^3+y\rangle\cdot\mathrm d\vec r$

By Green's theorem, this is equivalent to the double integral,

$\displaystyle\iint_D\left(\frac{\partial(4x^3+y)}{\partial x}-\frac{\partial(2xy^2)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=\iint_D(12x^2-4xy)\,\mathrm dx\,\mathrm dy$

where $D$ is the region with boundary $C$. This integral is equal to

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