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I have to verify Green's theorem for a question with the region being a parallelogram with vertices $(0,0), (1,1),(2,0),(3,1) $ and I'm having trouble with the standard approach of finding limits of integration with Fubini’s Theorem. Now here is where I'm having the problem.

I know that if I'm integrating first with respect to y and then with respect to x, I have to draw a vertical line cutting through the region R in the direction of increasing y and the upper and lower cuts mark the corresponding limits of y as functions of x. I have found x limits to be from 0 to 2 but I'm having a lot of trouble with limits of y.

It seems like the line enters at $y = 0$ and leaves at $y = 1 $. How do you write them as functions of x? Thank you.

I have uploaded a picture to show my attempt and where I got stuck. My attempt

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It is something like,

$$\int_{y=0}^{1}\int_{y}^{y+2} \left(...\right)dxdy$$

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