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Homework Help: Changing the variable in multiple integrals

  1. Jan 22, 2009 #1
    1. The problem statement, all variables and given/known data



    taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi).

    2. Relevant equations

    [tex]I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv[/tex]

    where J is the Jacobian.

    3. The attempt at a solution

    Okay so I've just been learning this for the first time, so I may be doing it completely wrong!

    I used the transformations u=x-y, v=x+y which give the Jacobian as 2.

    Now i wasn't sure how to get the new limits for the integrals. What I did was apply the transformation above to give new vertices:

    (pi,0) -> (pi,pi)
    (0,pi) -> (-pi,pi)
    (pi,2pi) -> (-pi,3pi)
    (2pi,pi) -> (pi,3pi)

    This gives a simple rectangle, so then i just wrote

    [tex]I = 2*\int^{3\pi}_{\pi}\int^{\pi}_{-\pi}u^2sin^2(v)dudv = \frac{4\pi^{4}}{3}.[/tex]

    I wish this was right, but i've a feeling it's not :-(

    Any help greatly appreciated!
    Last edited: Jan 22, 2009
  2. jcsd
  3. Jan 22, 2009 #2


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    The rectangle looks ok. But haven't you got the jacobian factor upside down?
  4. Jan 22, 2009 #3
    Ah yeah, should be 1/2. Other than that though does my method look correct?

  5. Jan 22, 2009 #4


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    Looks ok to me.
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