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The use of Fubini's theorem

  1. Nov 24, 2005 #1
    Hi, I usually dont have any problems with Fubini's theorem, but there is something I just cant figure out. Let f be integrable, and a some positive constant. How do i apply the theorem to this integral:
    [tex] \int_0^a\int_x^a \frac{1}{t}|f(t)|dtdx [/tex]
    Really; I know the answer is
    [tex]\int_0^a \int_0^t \frac{1}{t}|f(t)|dxdt[/tex]
    but I just dont get it. To me this is not obvious (should it be?). Can someone explain this to me?
     
  2. jcsd
  3. Nov 24, 2005 #2
    I guess I have made it more difficult then it really is. I just want to know why the second integration domain turns out like that. For simplicity put [tex]\frac{1}{t}|f(t)| = t[/tex] for example (and [tex]t\in (0,a)[/tex]). Then the integration area becomes
    [tex]\int_0^a\int_x^a t dtdx = \int_?^?\int_?^?t dxdt[/tex]
    Why?
     
  4. Nov 24, 2005 #3

    shmoe

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    Draw a picture in the x-t plane. Your region is bounded by the curves t=a, x=0 and x=t. For a fixed t, x ranges from 0 to t. t itself can range from 0 to a.
     
  5. Nov 24, 2005 #4
    Oh, man... I must be tired :) The original problem was not posted like this. I did not realize that the 2D integration area was infact the upper triangle of the square [0,1]^2. I thought I was dealing with the hole square... Stupid me :)

    Then of course it is easy.
    Thanks
     
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