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Math Help - T/F on integrals

  1. #1
    Junior Member BayernMunich's Avatar
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    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 :

    \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)
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by BayernMunich View Post
    True of False ? if T prove it, if F give a counter-example.

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

    \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

    f(x)=0
    Last edited by Ackbeet; May 16th 2011 at 08:12 AM.
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  3. #3
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    Take f(x) = sinx, g(x) = cosx, a = 0, b = π.

    EDIT: Looks like I over thought it too!
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  4. #4
    A Plied Mathematician
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    Quote Originally Posted by TheEmptySet View Post
    Don't over think the problem what if

    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.
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    Could be wrong, but is this a valid counter-example?
    I think you're right!
    Last edited by TheCoffeeMachine; May 16th 2011 at 08:43 AM.
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  6. #6
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    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!
    Last edited by TheCoffeeMachine; May 16th 2011 at 09:04 AM.
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  7. #7
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Here is what I was thinking

    Let f be a constant function then

    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

    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

    a=b-1

    and

    f(x)=c

    In either of these two cases equality holds so the statement is false.
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  8. #8
    A Plied Mathematician
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    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.
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