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?
Best Answer
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.