I need help on the process of setting up the limits of integration in this problem.
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.
$$\int_C (y + e^\sqrt x)\,dx + (2x + \cos y^2)\,dy,$$
where $C$ is the boundary of the region enclosed by the parabolas $y =x^2$ and $x = y^2$.
Ok So I know how to apply greens theorem but I'm not sure how to take the limits of integration on this type of problem. I know how to find where the two parabolas intersect, at $(0,0)$ and $(1,1)$, I got that by setting $y = x^2$ to $y = \sqrt x$ but I'm not sure what the process is in figuring out where to put them in the integrands. Can someone describe this in detail? I can't find a detailed description anywhere online.
I want to say the outer y boundaries are $0$ to $1$ and the inner x boundaries are $0$ to $y^2$ bt I'm not sure.
Green's Theorem states that
$$\int_CP\left(x,y\right)\,dx+Q\left(x,y\right)\,dy=\int_\Omega Q_x\left(x,y\right)-P_y\left(x,y\right)\,dA,$$
where $\Omega$ is the region enclosed by the closed curve $C$, which is oriented counterclockwise.
Therefore, your problem can be reformulated as
$$\int_0^1\int_{x^2}^{\sqrt x}\,dy\,dx.$$
I will leave the details to you.
Edit: Figuring out the limits of your integrals is a cal 1 problem. Just draw what the thing looks like:
I treat it as a type 1 region, and it is clear that $0\leq x\leq1$ and that $x^2\leq y\leq\sqrt x$.