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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?

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    $\begingroup$ If you continue with mathematics and enter into studying non-Euclidean geometry or objects of various and sundry strange topologies, I guess the concerns you mention might become important. And even at the level where you are now, there might be sometimes where hypothesizing a "fictional" fourth dimension makes some calculations more convenient somehow. But, no, at the level where you are right now, a triple integral is simply a triple integral, no further context needed. If you know the function and the shape of the volume, that is all there is to be said. $\endgroup$ – bob.sacamento Dec 10 '18 at 16:03
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    $\begingroup$ Possible duplicate of What does a triple integral represent? $\endgroup$ – BigbearZzz Dec 10 '18 at 16:04
  • $\begingroup$ A single integral can give the area under a curve, but often that's not what the integral really means. For example, a single integral can give the height of a ball after it has fallen for some time with an accelerating downward speed. We can make a graph of the speed over time and use "area under the curve" to help find the ball's height, but the height is not actually area under a curve and we don't really need to think about any areas in order to compute the height. $\endgroup$ – David K Dec 10 '18 at 22:02
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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.

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