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Homework Help: Square of an integral

  1. Jul 6, 2008 #1
    1. The problem statement, all variables and given/known data
    What theorem do you use to prove that

    [tex]\left(\int_a^b f(x) dx \right)^2 = \int_a^b f(x) f(y) dx dy[/tex]

    ?


    2. Relevant equations



    3. The attempt at a solution
     
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  3. Jul 6, 2008 #2

    morphism

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    What exactly does [itex]\int_a^b f(x) f(y) dx dy[/itex] mean?
     
  4. Jul 6, 2008 #3

    Defennder

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    I don't think you need to use any theorem. Just look at what the RHS means, as morphism said.
     
  5. Jul 6, 2008 #4

    arildno

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    You most certainly need a theorem here!
    It is called Fubini's theorem.

    The essence is that double integrals CAN be handled as iterated integrals, simplifying our job immensely.
     
  6. Jul 6, 2008 #5
    Sorry I meant [tex]\int_a^b \int_a^b f(x) f(y) dx dy[/tex].

    But how do you prove the LHS is a double integral OR an iterated integral?
     
  7. Jul 6, 2008 #6

    HallsofIvy

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    Fubini's theorem states that the double integral
    [itex]\int \int F(x,y) dx dy[/itex]
    is the same as the repeated integral
    [tex]\int \left(\int F(x,y)dy\right) dx[/itex]
    where the "inner integral" is taken treating x as a constant.

    The crucial point here is that your F(x,y)= f(x)f(y) is a product of two functions, one a function of x only, the other a function of y only.
    [tex]\int\left(\int f(x)f(y)dy\right)dx= \int f(x)\left(\int f(y)dy\right) dx[/tex]
     
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