$\int_{C} xydx + x^2dy$, where C is the rectangle with vertices $(0,0),(3,0),(3,1),(0,1)$.
I don't understand why the result does not agree with the predict by Green theorem: The rectangle has four regions, two lines parallel to x, and so is to y.
The first line $(0,0);(3,0)$ the integral is zero, since $y = dy = 0$
The second line $(3,0);(3,1)$ is $\int 9 \space dy = 9$
The third line $(3,1);(0,1)$ is $\int x \space dx = 0-9^2/2$
The fourth line $(0,1);(0,0)$ is zero
So the results is $9-9²/2$
Now, the green theorem: $\int (2x-x)dxdy = 9^2/2$
?? This is different from the predict!