Before we get started lets note a few things. is bounded above on by 1. There is an irrational and rational point in every interval of postive length.

Let Pick an such that .

Now lets construct a partion of [0,1]. let n > N

let

Now set

So f is integrable on [0,1]. Since it is integrable the value of the integral is equal to the supreemum of the lower integrals. Since it is always zero the value of the integral is 0.

Good luck.