Integration by change of variable

[asif zaidi]

Hello:

In Problem Solution part below, I am not sure of Step 2 and am having problems with Step 3.
Thanks in advance


Problem Statement:

Let D = {(x1,x2)| x1,x2>0, 1<= x1^{2} - x2^{2} <=9, 2 <= x1x2 <= 4.
Use hyperbolic coordinates g(u,v) = {u^{2} - v^{2}, uv} to show that

integ on D of (x1^{2} + x2^{2}) dx1dx2 = 8 ---- equation 1.


Problem Solution:

Step1: To show that Jacobian of g is non-zero
This was OK. Det is 2u^{2} + 2v^{2}. it is also given that u,v >0. Therefore det will always be >0


Step2: To convert x1,x2 into u, v
Substituting (u^{2} - v^{2})^{2} into x1^{2} and (uv)^{2} into x2^{2} equation 1 changes to after manipulation to ( I also did multiply by the Jacobian)

integ on R of (2u^{6} + 2v^{6}

Step3: To determine limits of u,v.
This is where I am having problems with

How do I translate 1<= x1^2 - x2^2 <=9 and 2 <= x1x2 <= 4 into u, v. Everything I do doesn't give me the ans in the question.



Thanks

Asif

 

[HallsofIvy]

You are taking x1= u²- v² and x2= uv?

It is far simpler to let u= x1²- x2² and v= x1x2.

 

[asif zaidi]

Excellent - thank you.
It worked.