# 5.4 Triple integrals

 Page 1 / 8
• 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] . A rectangular box in ℝ 3 divided into subboxes by planes parallel to the coordinate planes.

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.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By By       By By