Change of Variables: Evaluating Double Integral over R with Cosine Function

[fk378]

Homework Statement

Evaluate the double integral over R of cos[(y-x)/(y+x)] dA where R is the trapezoidal region with vertices (1,0) (2,0) (0,1) and (0,2).

The Attempt at a Solution

First I set u=y-x, v=y+x. I have 4 sides in the xy-plane that need to be transformed into the uv-plane. Side 1 is y=-x+2, Side 2 is y=-x+1, and then Side 3 & 4 are 0<x<2 (where the less than sign signifies less than or equal to).

I solved for v, and got v=1, v=2, so I have those limits. But I can't figure out how to find the limits for u.

 

[rootX]

u+v = 2y
u-v = -2x

using x = 0, y =y and x =x, y = 0

you can get u = v and u = -v

I don't know better method to find these limits