How to find limits of integration in changed coordinates

Question

Given integral = $\iint_R \frac{x-y}{(x+y)^3}dxdy$ :$R=[0,1]\times[0,1]$

Let $u = x+y, v= x-y \implies x = \frac{u+v}{2}, y = \frac{u-v}{2}$

Now x limits are from 0 to 1 and y limits are from 0 to 1. I am not able to understand how to find limits of integration in terms of u, v..

Pls enlighten me.

Answer 1

We need to consider the original domain in x-y plane which is a square $[0,1]\times[0,1]$ and the consider on that plane the lines

• $u = x+y$
• $v= x-y$

to see that the range for the two new variables is

• $0\le u\le 2$
• $-1\le v\le 1$

but note that in this case the two variables are not independent thus we need to fix the limits of variation for a first variable and the find the range for the second that is for example

• $0\le u\le 1$
• $-u\le v\le u$

and

• $1\le u\le 2$
• $u-2\le v\le 2-u$