Evaluate line integral using green's theorem

Question

I'm having a little problem understanding this question on Green's Theorem. It asks to use Green’s Theorem to evaluate the line integral $\int_C F.dr$, where $F=(e^{y^2} −2y)i+(2xye^{y^2} +\sin(y^2))j$ and C goes along a straight line from $(0,0)$ to $(1,2)$ and continues along a straight line to $(3, 0)$.

I tried using the circulation form of Green's Theorem and ended up with the answer 4+2 = 6. I then tried using the divergence form and seemed to end up with a different answer. Can I check if this is the correct way of doing it? Am I supposed to get the same answer whether I'm using circulation or divergence form? The correct answer is -3.

Would really appreciate any help!

Answer 1

There are a few problems that I think you're running into:

1. Green's theorem assumes counterclockwise orientation along the path. The path you've described is clockwise, so it should be $-6$ instead of $6$.
2. Green's theorem is for closed paths. The curve you described doesn't include the bottom portion of the enclosed area, i.e., the line from $(3,0)$ back to the origin. Note that $y=0$ along this line (which I'll call $C_0$), so a quick computation shows that $F(x,0) = \mathbf{i}$, since $e^{0^2}=1$ is the only nonzero term along $C_0$. Thus $$\int_{C_0} \mathbf{F}\cdot d\mathbf{r} = \int_3^0 1\,dx = -3.$$ Subtracting this from the value from the closed path from before, we have $-6-(-3) =-3$.