1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    Looks ok to me.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Changing the variable in multiple integrals