Double and triple integrals - does the order of integration matter? What does f(x,y,z) "do"? 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.
 A: 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.
A: Yes, you can define the order of integration as you want of dxdydz you just have to change the limit of integration.
secondly, here in your case f(x,y,z) is mass per unit volume
in general it is density function
