# Homework Help: Help with Change of variables and evaluating area?

1. Mar 7, 2012

### Suy

1. The problem statement, all variables and given/known data

Let I=∫∫D (x2−y2)dxdy, where
D=(x,y): {1≤xy≤2, 0≤x−y≤6, x≥0, y≥0}
Show that the mapping u=xy, v=x−y maps D to the rectangle R=[1,2]χ[0,6].

(a) Compute $\frac{\partial(x,y)}{\partial(u,v)}$ by first computing $\frac{\partial(u,v)}{\partial(x,y)}$.

(b) Use the Change of Variables Formula to show that I is equal to the integral of f(u,v)=v over R and evaluate.

2. Relevant equations

3. The attempt at a solution

(a) $\frac{\partial(u,v)}{\partial(x,y)}$=|-(y+x)|
so, $\frac{\partial(x,y)}{\partial(u,v)}$=$\frac{1}{y+x}$