Use Green's Theorem to evaluate $\mathbf{F}(x,y)=\langle y^{2}\cos x, x^{2}+2y\sin x\rangle$, where $C$ is the triangle from $(0,0)$ to $(2,6)$ to $(2,0)$ to $(0,0)$.
Taking the appropriate partial derivatives, I have my integral set up as $\displaystyle \int_{C}y^{2}\cos x\, dx +(x^{2}+2y\sin x)\,dy = \displaystyle \int_{2}^{0}\int_{0}^{3x}2x\, dy\, dx$. My reasoning for the order of the limits of integration was that since we are going from 2 to 0 along the bottom edge of the triangle, the $dx$ integral should have limits of integration from 2 to 0.
For the $dy$, integral, however, I'm not quite as sure. In the y-direction, it seems as though we're going from 0 to $3x$, because of how the curve is oriented along the hypotenuse.
But, I'm second-guessing myself - I still don't feel like I have the hang of setting these problems up, so if you could 1) tell me whether I'm right, and 2) if I'm not right, explain to me why, so I don't make the same mistake again.