# Thread: T/F on integrals

1. ## T/F on integrals

True of False ? if T prove it, if F give a counter-example.

Suppose that f & g are continuous functions on [a,b], then :

$\displaystyle \int_a^b f(x) \cdot g(x) \, dx \neq \left( \int_a^b f(x) \, dx \right) \cdot \left( \int_a^b g(x) \, dx \right)$

2. Originally Posted by BayernMunich
True of False ? if T prove it, if F give a counter-example.

Suppose that f & g are continuous functions on [a,b], then :

$\displaystyle \int_a^b f(x) \cdot g(x) \, dx \neq \left( \int_a^b f(x) \, dx \right) \cdot \left( \int_a^b g(x) \, dx \right)$
Don't over think the problem what if

$\displaystyle f(x)=0$

3. Take f(x) = sinx, g(x) = cosx, a = 0, b = π.

EDIT: Looks like I over thought it too!

4. Originally Posted by TheEmptySet
Don't over think the problem what if

$\displaystyle f(x)=0$
Could be wrong, but is this a valid counter-example? The statement is, I think, false in general, but it's true with this example, isn't it? I'd go with TCM's example, or you could do something even simpler with f(x) = g(x) = 2, and a = 0, b = 2.

5. Originally Posted by Ackbeet
Could be wrong, but is this a valid counter-example?
I think you're right!

6. Wait: If h(x) = k for all x then h(b)-h(a) = k-k = 0?
This makes LHS = RHS so the counterexample holds!
Or perhaps I'm missing something, as I'm so often!

7. Here is what I was thinking

Let f be a constant function then

$\displaystyle c\int_{a}^{b}g(x)dx=\frac{c}{b-a}\int_{a}^{b}g(x)dx$

I just want to show that this equality can hold so I get

$\displaystyle c\left(1-\frac{1}{b-a} \right)\int_{a}^{b}g(x)dx=0$

This gives two possible cases

either c=0 and a and b don't matter or

$\displaystyle a=b-1$

and

$\displaystyle f(x)=c$

In either of these two cases equality holds so the statement is false.

8. Hmph. Chalk another one down to a mis-placed "not" in my brain. I knew in my head that the integrals were not, in general, equal. That means that the statement "These two integrals are not equal, in general" is itself true. There are situations when the integrals are equal, and other situations when they're not equal.