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Proof f(x)=0 when integral from a->b equals zero

  1. Jul 30, 2010 #1
    Hello all,
    suppose f is a continuous non negative function if int(f(x),x=a..b)=0 show that f(x)=0

    what i have done is used mean value theorem to show some point c is such that f(c)=0. from here though i can only think of a verbal argument (since f is non negative) to explain why f(x)=0.

    i am wondering if there us an obvious use of the fundamental theorems here that im not seeing, or just some simple method.

    thank you for any help
  2. jcsd
  3. Jul 30, 2010 #2


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    Straight from the definition of an integral. If there's a point where f(x) is non-zero, can you prove there must exist some small amount of area underneath the graph around that point?
  4. Jul 30, 2010 #3
    you mean darboux or reimann integral? i feel like this might be easier to solve then using sums. perhaps it can be shown the indeterminate integral is a constant function?
  5. Jul 30, 2010 #4
    I think Darboux is a little bit more comfortable here. Just construct a partition such that in at leas one of the intervals f is greater than some fixed [tex]\delta > 0[/tex].
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