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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[tex]^{2}[/tex] - x2[tex]^{2}[/tex] <=9, 2 <= x1x2 <= 4.

Use hyperbolic coordinates g(u,v) = {u[tex]^{2}[/tex] - v[tex]^{2}[/tex], uv} to show that

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

Problem Solution:

Step1: To show that Jacobian of g is non-zero

This was OK. Det is 2u[tex]^{2}[/tex] + 2v[tex]^{2}[/tex]. it is also given that u,v >0. Therefore det will always be >0

Step2: To convert x1,x2 into u, v

Substituting (u[tex]^{2}[/tex] - v[tex]^{2}[/tex])[tex]^{2}[/tex] into x1[tex]^{2}[/tex] and (uv)[tex]^{2}[/tex] into x2[tex]^{2}[/tex] equation 1 changes to after manipulation to ( I also did multiply by the Jacobian)

integ on R of (2u[tex]^{6}[/tex] + 2v[tex]^{6}[/tex]

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

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

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Integration by change of variable

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