# How to find limits of integration of a parallelogram with these vertices?

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

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