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I recently had asked this question that what would be $\int_0^1\int_0^1\int_0^1\left(\lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor\right) \, dx\,dy \,dz$ and was pretty convinced with the answers saying it's $0$ which was what I guessed too. At that time I didn't had the key to this question and now I do but as it turns out, it's $3$. I'm quite confused right now

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    The integrand function is $0$ in the interior of $[0,1]^3$. Hence, the integral is $0$. (I am supposing $[t]$ is the integer part of $t$: in such a case, for $t \in (0,1)$ we have $[t]=0$). – 2017-02-28
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    Are you sure it is the floor function? E.g. if it is the ceiling function the integral is indeed 3. – 2017-02-28

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