The following question is based on multi-variable calculus and real analysis.

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> A set $A\subset [0,1]$ is dense in $[0,1]$ **iff** every open interval that intersects [0,1] contains a point of A. Suppose $f:[0,1]\to \mathbb{R}$ is integrable and $f(x)=0,\enspace \forall \enspace x \in A.$ Also assume that $f(x)\geq 0,\enspace \forall \enspace x \in [0,1].\enspace$Show that $\int\_0^1f(x)dx=0$
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I have no idea how to prove the above stuff rigorously. Then only thing I can think of 'if every open interval that intersects $[0,1]$ contains a point of A' is that set A contains all points of the interval $[0,1]$. Being that said it would be obvious for the integral to be zero.  

This is all I could think of. I am pretty sure that my assumption would be incorrect.