Question
For a real number $x$,Then
$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left(\left[x\right]+\left[y\right]+\left[z\right]\right)dxdydz$
$[$ $]=$greatest integer function
MY approach
MY first thought is that answer can be $0$, because limit is $0$ to $1$. But it is wrong at the same time because at $(1,1,1)$, grt function will not give $0$
I tried to apply Dirichlet Triple Integral Theorem But it does not Fit here properly
Based ON all comments
$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left(\left[x\right]+\left[y\right]+\left[z\right]\right)$dxdydz= 3$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}$dxdydz=3
$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \lceil x \rceil + \lceil y\rceil + \lceil z\rceil dxdydz=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} 3 dxdydz=3.$$
Note that on the interior of the region of integration the integrand $$\lceil x \rceil + \lceil y\rceil + \lceil z\rceil =3$$
Thus the value of the integral is the volume of the region times $3$.