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Help with Change of variables and evaluating area?

  1. Mar 7, 2012 #1

    Suy

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

    Let I=∫∫D (x2−y2)dxdy, where
    D=(x,y): {1≤xy≤2, 0≤x−y≤6, x≥0, y≥0}
    Show that the mapping u=xy, v=x−y maps D to the rectangle R=[1,2]χ[0,6].

    (a) Compute [itex]\frac{\partial(x,y)}{\partial(u,v)}[/itex] by first computing [itex]\frac{\partial(u,v)}{\partial(x,y)}[/itex].

    (b) Use the Change of Variables Formula to show that I is equal to the integral of f(u,v)=v over R and evaluate.

    2. Relevant equations

    3. The attempt at a solution

    (a) [itex]\frac{\partial(u,v)}{\partial(x,y)}[/itex]=|-(y+x)|
    so, [itex]\frac{\partial(x,y)}{\partial(u,v)}[/itex]=[itex]\frac{1}{y+x}[/itex]

    (b)I have to evaluate I, but I have no idea how, please help!!
     
  2. jcsd
  3. Mar 7, 2012 #2
    The change of variable converts the irregularly shaped domain D into the rectangle R, where you can evaluate I easier, to evaluate I perform the suggested change of variable, substitute in the jacobian, and evaluate over R, the jacobian you evaluated is correct, except it should be in terms of u, v, not x, y.
     
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