Mass of a three dimensional region

There's absolutely no reason (other than sanity) to stop at double integrals. Suppose we have a three dimensional object with a mass density function given $\delta (x,y,z)$. If the volume of the object occupies a region W of three-space, then

$$\mbox{mass} = \int_W \delta(x,y,z) dV.$$

The integration is with respect to the infinitesimal volume element,

$$dV = dx dy dz \quad (\mbox{in Cartesian coordinates}.)$$

This integral (a triple integral) is an iterated integral as well. It consists of three integrations, one in x, one in y and one in z.

If the region W is given by $0 \le x \le 1, 0 \le y \le 2, 0 \le z \le 4$and the density function is $\delta(x,y,z) = 3 + z$, then the mass is

$$\int_W \delta dV = \int_0^4 \int_0^2 \int_0^1 \delta_x,y,z) dx dy dz$$

$$= \int_0^4 \int_0^2 \int_0^1 (3+z) dx dy dz$$

$$= \int_0^4 \int_0^2 (3x + zx)_0^1 dy dz$$

$$= \int_0^4 \int_0^2 (3 + z) dy dz$$

$$= \int_0^4 (3y + zy)_0^2 dz$$

$$= \int_0^4 (6 + 2z) dz = (6z + z^2)_0^4 = 40.$$