Changing the variable in multiple integrals

  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. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

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

  5. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    Looks ok to me.
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