How do I calculate this double integral using a change of variables?

[mld993]

The problem is as follows.

Calculate the double integral of cos ((x-y)/(x+y)) dA over R, where R is the triangle bounded by the points (0,0), (2,2), and (2 + pi, 2 - pi).

I understand that you have to set U = x-y and V = x+y. However, I am having a hard time finding the bounds on the integral because the slope of the lines isn't 1. I can't simply add x to the other side of each equation because of the 2+pi and 2-pi. Any help would be appreciated. Thanks!

 

[CAF123]

The problem is as follows.

Calculate the double integral of cos ((x-y)/(x+y)) dA over R, where R is the triangle bounded by the points (0,0), (2,2), and (2 + pi, 2 - pi).

I understand that you have to set U = x-y and V = x+y. However, I am having a hard time finding the bounds on the integral because the slope of the lines isn't 1. I can't simply add x to the other side of each equation because of the 2+pi and 2-pi. Any help would be appreciated. Thanks!

It is a case of mapping the coordinates of the points in Cartesians to uv space. When x=y=0, what are u and v? When x=y=2, what are u and v? Similarly, for x,y=2±π. Once you have pairs (u,v), you can sketch the resulting region and then determine the bounds for u and v.