The line integral you need to compute is
[tex]\displaystyle\int_C\langle2xy^2,4x^3+y\rangle\cdot\mathrm d\vec r[/tex]
By Green's theorem, this is equivalent to the double integral,
[tex]\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[/tex]
where [tex]D[/tex] is the region with boundary [tex]C[/tex]. This integral is equal to
[tex]\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}[/tex]
