TITLE: Finding volumes — when to use double integrals and triple integrals?
QUESTION [7 upvotes]: This is not a technical question at all, but I'm quite confused about what should I use to compute volumes in $\mathbb{R}^3$ with integration.
I've read somewhere that a double integral gets the volume swapping across the $x$ and $y$ axis while a triple integral just integrate the whole thing at once, how accurate is this? Can a volume expressed by a double integral be expressed by a triple integral?, And can a triple always be expressed by a double? This one doesn't seem true, but I don't have a good answer to why.
I also found this comments while reading openstudy.com made by someone named  KingGeorge one year ago:

For a double integral you have to integrate some function, for a triple integral, you integrate 1.

Does this mean that using an integral to get a volume always should look like $\iiint  dxdydz$ without any function?

Geometrically, there are a few things you can be looking at. One, you're finding a 4-volume. That is, the 4-dimensional equivalent of volume. Two, if the volume in the region you're integrating in has a changing density, you could be finding the total mass.

I'm not sure if I understand correctly, but this means that a triple integral does not compute exactly the volume I want but a 4-D equivalent?.

REPLY [2 votes]: Both double and triple integrals can be used to calculate volumes of three dimensional objects. For triple integration, you can reduce the triple integral into a double integral by first calculating the Z component (or any component depending on the "type" of object), and then calculating the double integral over the remaining 2D region. Triple integration is used for functions of three variables and double integrals are used for functions of two variables. For a function of 1 variable, you use the methods of calculus 1 to rotate the objects (solids of revolution) about an axes or line.