Change of variables in a double integral

[tjackson3]

Homework Statement

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 $\rho(x,y) = x^2 + y^2.$

Homework Equations

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?

 

[SammyS]

Homework Statement

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 $\rho(x,y) = x^2 + y^2.$

Homework Equations

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?

Why do you want to solve for x & y ?

 

[tjackson3]

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

 

[SammyS]

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

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

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

Does that help ?

 

[tjackson3]

Very much. Thank you!

 

[SammyS]

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.

 

[tjackson3]

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