And this is the answer they have given us:

They converted to polar coordinates. I've seen this type of problem being solved without conversion to polar coordinates. I tried to evaluate the integral without converting to polar and I ended up with a different answer:

$\displaystyle \int^1_0 \int^{-y+1}_{x=0} 2x-2y dx dy$

$\displaystyle \int^1_0 y^2-2y+1dy = \frac{1}{3}-1+1=\frac{1}{3}$

But my answer does not agree with the model answer! Shouldn't we end up with the same result regardless of whether we use polar or rectangular coordinates? I appreciate any help because I have an exam tomorrow...