(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the mass of the plane region R in the first quadrant of the (x,y)-plane bounded by the hyperbolas

[itex] xy=1 \,\,\,\,\,\,\,\,\,\, xy=2\,\,\,\,\,\,\,\,\,\, x^2-y^2=3\,\,\,\,\,\,\,\,\,\, x^2-y^2=5 [/itex]

Assume the density at the point (x,y) is [itex] \rho=x^2+y^2 [/itex]

2. Relevant equations

[tex] m=\int \int_R \rho(x,y)dxdy [/tex]

3. The attempt at a solution

I am stuck at finding a suitable change of variables to transform this into a "nice" region so I don't have to perform 3 seperate integrals. Even if I took the long way (3 integrals) the point of intersection is not easy to find analytically. What is a clever change of variables that I can use?

I have tried the following:

[itex] u=xy \,\,\,\,\,\,\,\,\,\, v=x^2-y^2 [/itex]

then I can't find a nice expression for [itex] \rho(u,v) [/itex]

I also tried

[itex] x=u/v \,\,\,\,\,\,\,\,\,\, y=v [/itex]

but then solving for v is ugly.

I even tried

[itex] u=x^2 \,\,\,\,\,\,\,\,\,\, v=y^2 [/itex]

which gave another ugly region.

Please help, thank you.

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# Mass in a plane region

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