Square of an integral

1. Jul 6, 2008

ehrenfest

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

$$\left(\int_a^b f(x) dx \right)^2 = \int_a^b f(x) f(y) dx dy$$

?

2. Relevant equations

3. The attempt at a solution

2. Jul 6, 2008

morphism

What exactly does $\int_a^b f(x) f(y) dx dy$ mean?

3. Jul 6, 2008

Defennder

I don't think you need to use any theorem. Just look at what the RHS means, as morphism said.

4. Jul 6, 2008

arildno

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.

5. Jul 6, 2008

ehrenfest

Sorry I meant $$\int_a^b \int_a^b f(x) f(y) dx dy$$.

But how do you prove the LHS is a double integral OR an iterated integral?

6. Jul 6, 2008

HallsofIvy

Staff Emeritus
Fubini's theorem states that the double integral
$\int \int F(x,y) dx dy$
is the same as the repeated integral
$$\int \left(\int F(x,y)dy\right) dx[/itex] where the "inner integral" is taken treating x as a constant. The crucial point here is that your F(x,y)= f(x)f(y) is a product of two functions, one a function of x only, the other a function of y only. [tex]\int\left(\int f(x)f(y)dy\right)dx= \int f(x)\left(\int f(y)dy\right) dx$$

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