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?.
Best Answer
You can use both double and triple integrals when calculating a volume. Let me explain you using an example for calculating an area, same applies to volume.
Say you are looking to find the area under a curve $f(x)>0$ over the domain of integration. You must have learnt that : $$A = \int^a_bf(x)dx$$
What you are doing is basically summing infinitely many stripes of length $f(x)$ and base length $dx$.
Now observe that the following are the same : $$\int^a_b\int_0^{f(x)} dtdx = \int^a_bf(x)dx = A$$
So to calculate the area I can use both single and double integral. The second one is simply a shortcut. The first integral sums infinitely many little square of dimension $dt\times dx$ within the specified bounds for $t$ and $x$.
Similarly to find volumes : $$\int\int\int^{f(x,y)}_0dtdxdy = \int\int f(x,y)dxdy$$
The only difference is that the triple integral is a more basic approach in the sense that you really do it small cube by small cube. The double integral is a shortcut.