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Long Question

  1. Mar 17, 2008 #1
    1. Find the area of the region bounded by x^2 - xy + y^2 = 2:
    a)let x = au + bv, y= au - bv therefore, 3b^2v^2 + a^2u^2 = 2
    b) Choose a and b such that u^2 + v^2 = 1, therefore, a = sqrt 2 & b = (sqrt 6)/3

    c) Applying these results and changing variables into u and v, evaluate the integral //(x^2 - xy + y^2) dxdy, where the integral is bounded by the equation x^2 - xy + y^2 = 2.

    For the part c) I have found the J(u,v) = 4(sqrt 3)/3, but in the examples I have I am supposed to follow this up with an integral and I am not sure what to do next. Are there any suggestions?
     
  2. jcsd
  3. Mar 18, 2008 #2

    HallsofIvy

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    Integrate
    [tex]\int\int J(u,v)dudv= \int_{u=-1}^1\int_{v= -\sqrt{2- 2u^2}/\sqrt{6}}^{\sqrt{2- 2u^2}/\sqrt{6}}\[/tex]
    over the uv-ellipse. Isn't that why you found J(u,v)?
     
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