By Green's theorem, the circulation of [tex]\vec F(x,y)=\langle2xy^2,4x^3+y\rangle[/tex] around [tex]C[/tex] is equal to
[tex]\displaystyle\iint_D\frac{\partial(4x^3+y)}{\partial x}-\frac{\partial(2xy^2)}{\partial y}\,\mathrm dx\,\mathrm dy[/tex]
where [tex]D[/tex] is the region with [tex]C[/tex] as its boundary. The integral is equivalent to
[tex]\displaystyle\int_0^\pi\int_0^{\sin x}(12x^2-4xy)\,\mathrm dy\,\mathrm dx=\boxed{\frac{23\pi^2}2-48}[/tex]
