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Homework Help: Mass in a plane region

  1. Dec 24, 2011 #1
    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.
     
    Last edited: Dec 24, 2011
  2. jcsd
  3. Dec 24, 2011 #2

    vela

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    Your first set of transformations is the one you want. Now consider ##4 u^2+v^2##.
     
  4. Dec 24, 2011 #3
    Im still lost. I want [itex]\rho=x^2+y^2[/itex].
    [itex]4u^2=x^2+y^2[/itex]
    [itex]v^2=x^4-2x^2y^2+y^4[/itex]

    I'm still getting nowhere.
     
  5. Dec 25, 2011 #4

    vela

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    How did you get 4u2=x2+y2 from u=xy?
     
  6. Dec 25, 2011 #5

    SammyS

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    Isn't [itex]4u^2=4(xy)^2=4x^2y^2\,?[/itex]

    Now, add that to [itex]x^4-2x^2y^2+y^4\,?[/itex]

    Factor that !
     
  7. Dec 25, 2011 #6
    Sorry, I made a typo since I was copy pasting

    [itex] 4u^2=4x^2 y^2 [/itex]
     
  8. Dec 25, 2011 #7
    Ok, round 2, here it goes.

    [itex] 4u^2=4x^2 y^2 [/itex]
    [itex] v^2=x^4-2x^2y^2+y^4 [/itex]
    [itex] 4u^2+v^2=x^4+2x^2 y^2+y^4=(x^2+y^2)^2=\rho^2 [/itex]

    Thank you!
     
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