Proving the Square of an Integral Using a Theorem

[ehrenfest]

Homework Statement

What theorem do you use to prove that

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

?

Homework Equations

The Attempt at a Solution

 

[morphism]

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

 

[Defennder]

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

 

[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.

 

[ehrenfest]

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

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

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.

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

 

[HallsofIvy]

Fubini's theorem states that the double integral
$\int \int F(x,y) dx dy$
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.
$$\int\left(\int f(x)f(y)dy\right)dx= \int f(x)\left(\int f(y)dy\right) dx$$