Use Green’s Theorem to evaluate $\int_C <\sqrt{x}+y^3, x^2+\sqrt{y}>\cdot dr $

Question

Use Green’s Theorem to evaluate

$\int_CF\cdot dr$, if $F(x,y)=<\sqrt{x}+y^3, x^2+\sqrt{y}>$ and $C$ consists of the arc of the curve $y =sin(x)$ from $(0, 0)$ to $(\pi, 0)$ and the line segment from $(\pi, 0)$ to $(0, 0)$.

Trying to get the hang of the coding, so forgive if their are multiple edits. There is a photo tagged to the title if need be. Thanks

Answer 1

You should provide some of your attempts. Otherwise it is unclear where your confusion is. I assume that it is converting the line integral into a integral over a region. If your vector field is $F = (P,Q)$, then Green's Theorem says that $$ \int_C F\cdot dr = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \iint_D 2x - 3y^2 dA. $$