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Double integral change of variables

  1. May 31, 2014 #1
    1. The problem statement, all variables and given/known data

    Use the change of variables ##u=x+y## and ##y=uv## to solve:
    [tex] \int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx [/tex]
    2. Relevant equations

    3. The attempt at a solution

    So I got as far as:
    [tex] \int\int{}ue^vdvdu. [/tex]

    But I just can't find the region of integration in terms of ##u## and ##v##.
     
  2. jcsd
  3. May 31, 2014 #2

    CAF123

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    Gold Member

    What does the domain of integration look like in x-y space? Then consider what happens to points on the interior and boundary of that domain under the transformation to u-v space.
     
    Last edited: May 31, 2014
  4. May 31, 2014 #3
    the region looks like a triangle, basically the lower half of the unit square. But how do I transform this region into uv space? I tried to solve the equations of change of variables until I either got the bounds of ##u## as two numbers and the bounds of ##v## at least in terms of ##u##. Or the other way around.. I figured I could solve the integral either way.
     
  5. Jun 1, 2014 #4

    CAF123

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    Gold Member

    Yes, you can solve u=u(x,y) and v=v(x,y). Then choose points on the boundary of the domain in xy space and see what the corresponding point is in uv space. A point in the interior of the domain in xy space will point you towards the interior of the region in uv space.
     
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