# 5.4 Triple integrals

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• Recognize when a function of three variables is integrable over a rectangular box.
• Evaluate a triple integral by expressing it as an iterated integral.
• Recognize when a function of three variables is integrable over a closed and bounded region.
• Simplify a calculation by changing the order of integration of a triple integral.
• Calculate the average value of a function of three variables.

In Double Integrals over Rectangular Regions , we discussed the double integral of a function $f\left(x,y\right)$ of two variables over a rectangular region in the plane. In this section we define the triple integral of a function $f\left(x,y,z\right)$ of three variables over a rectangular solid box in space, ${ℝ}^{3}.$ Later in this section we extend the definition to more general regions in ${ℝ}^{3}.$

## Integrable functions of three variables

We can define a rectangular box $B$ in ${ℝ}^{3}$ as $B=\left\{\left(x,y,z\right)|a\le x\le b,c\le y\le d,e\le z\le f\right\}.$ We follow a similar procedure to what we did in Double Integrals over Rectangular Regions . We divide the interval $\left[a,b\right]$ into $l$ subintervals $\left[{x}_{i-1},{x}_{i}\right]$ of equal length $\text{Δ}x=\frac{{x}_{i}-{x}_{i-1}}{l},$ divide the interval $\left[c,d\right]$ into $m$ subintervals $\left[{y}_{i-1},{y}_{i}\right]$ of equal length $\text{Δ}y=\frac{{y}_{j}-{y}_{j-1}}{m},$ and divide the interval $\left[e,f\right]$ into $n$ subintervals $\left[{z}_{i-1},{z}_{i}\right]$ of equal length $\text{Δ}z=\frac{{z}_{k}-{z}_{k-1}}{n}.$ Then the rectangular box $B$ is subdivided into $lmn$ subboxes ${B}_{ijk}=\left[{x}_{i-1},{x}_{i}\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[{y}_{i-1},{y}_{i}\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[{z}_{i-1},{z}_{i}\right],$ as shown in [link] .

For each $i,j,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}k,$ consider a sample point $\left({x}_{ijk}^{*},{y}_{ijk}^{*},{z}_{ijk}^{*}\right)$ in each sub-box ${B}_{ijk}.$ We see that its volume is $\text{Δ}V=\text{Δ}x\text{Δ}y\text{Δ}z.$ Form the triple Riemann sum

$\sum _{i=1}^{l}\sum _{j=1}^{m}\sum _{k=1}^{n}f\left({x}_{ijk}^{*},{y}_{ijk}^{*},{z}_{ijk}^{*}\right)\text{Δ}x\text{Δ}y\text{Δ}z.$

We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.

## Definition

The triple integral    of a function $f\left(x,y,z\right)$ over a rectangular box $B$ is defined as

$\underset{l,m,n\to \infty }{\text{lim}}\sum _{i=1}^{l}\sum _{j=1}^{m}\sum _{k=1}^{n}f\left({x}_{ijk}^{*},{y}_{ijk}^{*},{z}_{ijk}^{*}\right)\text{Δ}x\text{Δ}y\text{Δ}z=\underset{B}{\iiint }f\left(x,y,z\right)dV$

if this limit exists.

When the triple integral exists on $B,$ the function $f\left(x,y,z\right)$ is said to be integrable on $B.$ Also, the triple integral exists if $f\left(x,y,z\right)$ is continuous on $B.$ Therefore, we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, $f$ is bounded on $B$ and continuous except possibly on the boundary of $B.$ The sample point $\left({x}_{ijk}^{*},{y}_{ijk}^{*},{z}_{ijk}^{*}\right)$ can be any point in the rectangular sub-box ${B}_{ijk}$ and all the properties of a double integral apply to a triple integral. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections.

Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s thereom for triple integrals exists.

## Fubini’s theorem for triple integrals

If $f\left(x,y,z\right)$ is continuous on a rectangular box $B=\left[a,b\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[c,d\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[e,f\right],$ then

$\underset{B}{\iiint }f\left(x,y,z\right)dV=\underset{e}{\overset{f}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{c}{\overset{d}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{a}{\overset{b}{\int }}f\left(x,y,z\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz.$

This integral is also equal to any of the other five possible orderings for the iterated triple integral.

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