The idea of splitting a path or line or area or surface into small pieces can be applied just as well to a volume.
We therefore define volume integrals by the same approach which by this time, I hope does not need repeating. (Here it is anyway: divide the volume into ever more pieces, say cubes, and let the diameter of the pieces go to zero, sum the contributions from each piece.)
Volumes are numbers rather than vectors in 3 dimensions, so the definition is quite straightforward.
When the integrand is 1, the integral becomes the volume itself.
A volume integral over V with density of whatever as integrand is the total amount of whatever that is in V. Such integrals are commonly encountered.
In particular, the volume integral of charge or mass density gives the charge or mass of in that volume.
You also encounter moments. The integral of r2 (with r representing distance to the z-axis) multiplied by the mass density over a volume V gives the moment of inertia about the z-axis of the material within V.
More generally, moments are integrals of powers of distance to an axis or point multiplied by appropriate densities. Moment of inertia is a "moment" about an axis.
Volume integrals are usually denoted as
Having defined all these entities we now turn to the question: how do we evaluate them?
The basic answer is: we use the fundamental theorem of calculus, which we will now describe, to either give us a direct answer or to reduce the integral in question to our performing one or a sequence of anti-derivatives.
Obviously the definitions here can be extended to dimensions beyond three.
In every case we can prove by the same argument that integrals of continuous functions over bounded closed regions always exist.