Of course, heuristically, a single integral gives area under a curve, and a double integral of a function gives the volume under the integrand and above a two-dimensional domain. Now, I understand that a triple integral of the number 1 gives the volume of the three-dimensional shape described by the limits of integration, but my professor told us that triple integrals are just integrals over "a 3D domain."
I suppose my confusion is this: does the value represented by a triple integral depend on the specific context of the problem, or are there different types of triple integrals that correspond to different meanings?
In some instances, one can use a triple integral to measure the volume of a $3D$ region, but triple integrals can also be used to find 'volume' between the graph of a $4D$ function and a $3D$ region.
Example:
Given a $3D$ region $E$, the volume of $E$, which we'll denote as $V(E)$, is given by $$V(E)=\iiint\limits_{E}\mathrm dx\mathrm dy\mathrm dz$$ But if you have a function $f(x,y,z)$, then you know that it's graph is going to be $4D$. But we can still find the 'volume' of $f$ over $E$: $$V(f,E)=\iiint\limits_{E}f(x,y,z)\mathrm dx\mathrm dy\mathrm dz$$
But at the end of the day a triple integral is just a triple integral. In some cases, thinking of the geometric meaning may make things more complicated.