I have not found anything about it, that's why I am asking it here.
As far as I know, the volume is: $$V=\iiint_E dxdydz$$ But does the order of dx, dy, dz matter? For example, if i can define the dy's limit with x and z, the dz's limit with x, and the dx's limit with constants, can i just simply write dydzdx?
And the other question: What does the f(x,y,z) "do" here? $$V=\iiint f(x,y,z) dxdydz$$ As I know, the mass can be calculated like this: $$m=\iiint \rho(x,y,z) dxdydz$$ But here the result is mass, not volume.
Your integrals are in first place integrals with respect to three-dimensional measure: $$\int_E1\>{\rm d}(x,y,z)\ ,\quad {\rm resp.},\quad\int_Ef(x,y,z)\>{\rm d}(x,y,z)\ ,$$ and they are limits of Riemannian sums of the following kind: $$\sum_{k=1}^N f({\bf r}_k)\>{\rm vol}(E_k)\ ,$$ whereby the $E_k$ form a partition of $E$ into almost disjoint "bricks" $E_k$.
Fubini's theorem tells us that such an integral can be written as a threefold nested integral, called a triple integral for short. If $E$ is convex such a triple integral looks like $$\int_a^b\int_{c(x)}^{d(x)}\int_{p(x,y)}^{q(x,y)}f(x,y,z)\>dz\>dy\>dx\ .\tag{1}$$ In order to determine the limits $c(x)$, $\ldots$, $q(x,y)$ appearing here you have to use the full and precise geometric description of $E$. The order of the nesting in $(1)$ is irrelevant, but the limits appearing in the integrals of course depend on the chosen order.