Calculate $$\displaystyle\oint_C(x\sin(e^y)+xy)dx+(\frac{x^2}{2}e^y\cos(e^y)+x^2y^3)dy$$ where $C$ is the polygon with vertices $(-1,0), (0,1), (1,1), (2,0), (0,-2) $ oriented counter clockwise.
I got $\displaystyle \int_{x=-1}^{x=1}\int_{y=x-2}^{y=1} (2xy^3-x)dydx$
Should I continue or am I wrong?
Your approach and application of Green's theorem is correct. However limits of integral is not correct. Please see shaded region in the diagram.
Integrating wrt $dx$ first will require two integrals.
For $ - 2 \leq y \leq 0$, $x$ is bound between lines $x = - \frac{y+2}{2}$ and $x = y + 2$.
For $0 \leq y \leq 1$, $x$ is bound between lines $x = y-1$ and $x = 2-y$.
So the integral should be,
$ \displaystyle \int_{-2}^0 \int_{-(y+2)/2}^{y+2} (2 xy^3 -x) ~ dx ~ dy ~ + $
$ \displaystyle \int_0^1 \int_{y-1}^{2-y} (2 xy^3 -x) ~ dx ~ dy $