Im not completely getting line integrals. Can you please help me with these questions.

I get 3 for (a) and (b) is that right?

Not sure how to do this one

I was thinking $\displaystyle \displaystyle\int_C F \cdot dr = \displaystyle\int_C F \cdot T ds = \displaystyle\int_C |F|cos \theta$

$\displaystyle |F| = \sqrt {x^4y^2 + 16}$. Not sure about theta. Is any of that the right thing to do?

I get $\displaystyle \displaystyle \frac{\partial f}{\partial y} = -e^xsiny$ and $\displaystyle \displaystyle \frac{\partial g}{\partial x} = -e^xcosy$. Not sure what to do next or if that is right.

For green's theorem it would be $\displaystyle \displaystyle\int \displaystyle\int_R (2x-2y) dA$? But im not sure about the limits. As for doing it directly, not sure about that either.

Please help. I really need to know this for my final exam