Triple integral

The triple integral extends the notion of single and double integration. The differential quantity is some volume $dV$.

The best interpretation involves that of density (mass, charge, etc.) or of concentration (of a chemical) within a given region. Alternatively, if $f=1$, we discuss some volume in ${\mathbb{R}}^3$. This is far more useful than double integrals, because computing volume with triple integrals isn’t the unsigned volume.

$${\iiint }_{D}f(x,y,z)dV=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f(x_k^{\ast },y_k^{\ast },z_k^{\ast })\Delta V_k$$

If $f$ is continuous, Fubini’s theorem over rectangular regions similarly holds. We can integrate in any order below!

$${\iiint }_{D}f(x,y,z)dV={\int }_a^b{\int }_c^d{\int }_e^{f}f(x,y,z)dzdydx$$

General regions

In the majority of cases, we don’t deal with simple rectangular regions and instead have a region of integration defined by functions. Our triple integral is usually expressed as:

$${\iiint }_{D}f(x,y,z)dV={\int }_{x=a}^{x=b}{\int }_{y=g(x)}^{y=f(x)}{\int }_{z=i(x,y)}^{z=h(x,y)}f(x,y,z)dzdydx$$

or some variation of this (maybe swapping around the order of integration). Take note of what we’re doing. The outermost integral bound is constant, the middle is a function of the outermost variable, and the innermost is a function of both variables.

Changing bounds

An exceptionally important idea is drawing the 3D region of integration, especially for complicated regions where there are scary-looking bounding functions. Draw the traces for each function and get a visual idea of what you’re working with.

A good way to think about how the bounds are changing are to build inequalities for each of the variables. For our outermost bound, we look only on the real line for that variable. For our middle bound, we look at the $xy$ (or whatever) plane. And for our outermost bound, we look at ${\mathbb{R}}^3$ as a whole.

An also important change is between rectangular and cylindrical or spherical coordinates.

Addendums

Formally, the idea of a triple integral refers to the 4D hypervolume of the region under the graph of $f$ over $R$, where $f\in {\mathbb{R}}^4$. We can’t visualise ${\mathbb{R}}^4$, so this geometrically means nothing to us and is frankly not really a useful interpretation. Which is why we use the above interpretations.

When calculating mass or charge (given a density and region), the integrand in our triple integral is the density function.