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Change of variables in a double integral

  1. Dec 19, 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 xy plane that is bounded by the hyperbolas [itex]xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5[/itex] where the density at the point x,y is [itex]\rho(x,y) = x^2 + y^2.[/itex]

    2. Relevant equations



    3. The attempt at a solution

    The region of integration lends itself to the change of variables [itex]u = xy, v = x^2-y^2.[/itex] 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?
     
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  3. Dec 19, 2011 #2

    SammyS

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    Why do you want to solve for x & y ?
     
  4. Dec 19, 2011 #3
    At the very least, I need to solve for [itex]x^2+y^2[/itex]
     
  5. Dec 19, 2011 #4

    SammyS

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

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

    Does that help ?
     
  6. Dec 19, 2011 #5
    Very much. Thank you!
     
  7. Dec 19, 2011 #6

    SammyS

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    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.
     
  8. Dec 19, 2011 #7
    Ah that is much nicer. Haha don't be modest now! Thanks again!
     
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