$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

16  Triple Integrals

Triple integrals follow a very similar general theory to double integrals: starting with a function \(f(x,y,z)\) on \(\RR^3\), we define the integral over a region \(E\) by breaking that region into small cubical volumes of size \(dV\) and building a 3-dimensional riemann sum. Taking the limit gives the triple integral, or

\[\iiint_E f\,dV\]

To evaluate such an expression, we need to break the integral into slices, and evaluate them one at a time. Such slicing relies on understanding the volume element in three dimensions, which is the volume of an infinitesimal box

\[dV = dxdydz\]

This lets us separate the triple integral into three consecutive integrals: first dx, then dy then dz. Or, because the order of multiplication doesn’t matter, we could do the integral in any of the other six possible orders \[dxdydz=dxdzdy=dydxdz\] \[=dydzdx=dzdxdy=dzdydx\]

16.1 Different Bounds:

16.1.1 Boxes

When the domain \(E\subset\RR^3\) is a coordinate box, described as \[E=\{(x,y,z)\mid a\leq x\leq b\, c\leq y\leq d\, e\leq z\leq f\}\]

This triple integral splits into an iterated integral with constant bounds:

\[\iiint_E F dV = \int_a^b\int_c^d\int_e^f F dzdydx\]

This integral could be done in any of the six possible orders, as all the bounds are constants, no order will be easier or harder than any other.

16.1.2 Variables in One Bound

If the domain \(E\) is described so that its bounds in at two of the variables are constants, and the third set of bounds are variables, then there is a preferred order in which to integrate. In particular, we know the final answer must be a number so we cannot have variables in the outermost set of bounds, and must be done earlier: the easiest situation is just to do it first!

For example, consider the following domain \(E\):

\[E=\{(x,y,z)\mid 0\leq x\leq 2, 0\leq y\leq 3, 0\leq z\leq x+y\}\]

Here the \(z\) bound is different depending on which point \((x,y)\) you are at, so we do the \(z\)-integral first. Since both the \(x\) and \(y\) bounds are constants

\[\iiint_E f\,d V = \int_0^2\int_0^3\int_0^{x+y}f\,dzdydx\]

16.1.3 Variables in Two Bounds

For more complicated domains, its possible that variables will appear in two of the bounds. (Because the final answer must be a number, we know the outer bounds must be constants, so they cannot appear in all three bounds).

In such cases, the innermost integral can have bounds depending on two variables (the next two to be integrated), and the middle integral can have bounds depending on the outermost integral. This way, at each stage the function only has variables left in it that are still going to be integrated away, and the result is a number. In this case, there is only one possible order in which the integral can be performed!

Here’s an example: if \(E\) is the following region

\[E=\{-y\leq x\leq yz,0\leq y\leq z+1,-1\leq z\leq 1\}\]

Then a triple integral must be performed with \(dx\) first, then \(dy\), and finally \(dz\):

\[\iiint_E f\,dV =\int_{-1}^1\int_0^{z+1}\int_{-y}^{yz}f\,dx dy dz\]

16.2 Describing the Bounds:

The thing that makes triple integrals challenging is not doing the integrals (its just three 1D integrals) or even choosing the order to do them in (as we saw above, once you have described the domain in terms of \(x,y,z\), its easy to decide which order to do the integral.) The difficult part is often just describing the bounds themselves!

This is mostly because visualizing 3D geometry takes some training to get used to! It’s helpful to look through many examples: please remember to be using the book (chapter 15), where each chapter is essentially just a giant list of example problems fully worked out! Additionally, Here is another collection of fully worked examples online:.

16.3 Video Resources