# Change of variables in a double integral

1. Dec 19, 2011

### tjackson3

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

Find the mass of the plane region R in the first quadrant of the xy plane that is bounded by the hyperbolas $xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5$ where the density at the point x,y is $\rho(x,y) = x^2 + y^2.$

2. Relevant equations

3. The attempt at a solution

The region of integration lends itself to the change of variables $u = xy, v = x^2-y^2.$ However, if I make this change of variables, it seems impossible to solve for x and y. Is there a better change of variables to make?

2. Dec 19, 2011

### SammyS

Staff Emeritus
Why do you want to solve for x & y ?

3. Dec 19, 2011

### tjackson3

At the very least, I need to solve for $x^2+y^2$

4. Dec 19, 2011

### SammyS

Staff Emeritus
Square v, that gives you x4 - 2x2y2 + y4

If you add 4x2y2 to that you will have x4 + 2x2y2 + y4 .

Does that help ?

5. Dec 19, 2011

### tjackson3

Very much. Thank you!

6. Dec 19, 2011

### SammyS

Staff Emeritus
By the way:

If you use the change of variables u=2xy, v=x2−y2 , then u2 + v2 = (x2 + y2)2 , which is a bit nicer.

The only reason I was able to help so quickly, was that I recently helped with a problem having a similar change of variable.

7. Dec 19, 2011

### tjackson3

Ah that is much nicer. Haha don't be modest now! Thanks again!