Line integral with Green theorem

Question

I will compute $\int_C \ e^xdx+xydy$ where C is the triangle with vertices (0,0), (1,1) and (0,2) with a positive orientation.

I started with $\iint (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy=\iint y \ dxdy$, then I integrate with respect to $x$ from $0$ to $1$ and $y$ from $y=x$ to $y=2$

But I get $\frac{11}{6}$ as the result which is wrong. Help me please.

Answer 1

Your application of Green's theorem is right but your upper bound of $y$ is not correct. $y$ is bound between $x$ and $2-x$. See the diagram.

So the integral should be,

$ \displaystyle \int_0^1 \int_x^{2-x} y ~ dy ~ dx$