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

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

## Answers and Replies

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Office_Shredder
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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?

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?

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 $$\delta > 0$$.