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Math Help - if \int a(x) b(x) dx = 0, is \int b(x) dx = 0 ?

  1. #1
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    if \int a(x) b(x) dx = 0, is \int b(x) dx = 0 ?

    Assume \int a(x) b(x) dx = 0 and a(x) \neq 0.
    Can this be simplified to \int b(x) dx = 0?

    Or in another way, if
     \int f(x) g(x) dx = \int f(x) h(x) dx ,
    then does this hold:
     \int g(x) dx = \int h(x) dx
    Last edited by dirkie; April 6th 2009 at 04:34 PM.
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  2. #2
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    Quote Originally Posted by dirkie View Post
    Assume \int a(x) b(x) dx = 0 and a(x) \neq 0.
    Can this be simplified to \int b(x) dx = 0?

    Or in another way, if
     \int f(x) g(x) dx = \int f(x) h(x) dx ,
    then does this hold:
     \int g(x) dx = \int h(x) dx
    No, they can't be simplified to that, unless a(x) and f(x) are constants.

    And in general, indefinite integrals aren't equal to zero as an arbitrary constant is always added to the result.

    If the integral was definite, there are still a variety of reasons why it would be equal to zero:

    If  \int_a^b f(x)g(x)dx = 0 , with  f(x) \neq 0 then some possibilities are:

     g(x) = 0

     a = b
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    Quote Originally Posted by Mush View Post
    No, they can't be simplified to that, unless a(x) and f(x) are constants.

    And in general, indefinite integrals aren't equal to zero as an arbitrary constant is always added to the result.

    If the integral was definite, there are still a variety of reasons why it would be equal to zero:

    If  \int_a^b f(x)g(x)dx = 0 , with  f(x) \neq 0 then some possibilities are:

     g(x) = 0

     a = b

    Thank you very much.

    If the integral was definite, i.e.  \int_a^b f(x)g(x)dx = 0

     g(x) = 0 is one solution, but not the only one, or is it?
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    Quote Originally Posted by dirkie View Post
    Thank you very much.

    If the integral was definite, i.e.  \int_a^b f(x)g(x)dx = 0

     g(x) = 0 is one solution, but not the only one, or is it?
    No. Like I said, the integral will also be zero if the limits are equal a = b.

    And it can also 'just happen'.
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  5. #5
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    Quote Originally Posted by dirkie View Post
    Thank you very much.

    If the integral was definite, i.e.  \int_a^b f(x)g(x)dx = 0

     g(x) = 0 is one solution, but not the only one, or is it?
    There are an infinite number of possiblilities.

    Here is one class of examples

    If the interval is symmetric about the origin (i.e. a=-b)then

    \int_{a}^{b}f(x)g(x)dx=0 if f is an even function and g is an odd function
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  6. #6
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    Quote Originally Posted by Mush View Post
    No. Like I said, the integral will also be zero if the limits are equal a = b.

    And it can also 'just happen'.
    Thanks. I am expecting the 'just happen' scenario for the problem, and was making sure if it exists. So it does.
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