# 5.4 Triple integrals  (Page 5/8)

 Page 5 / 8

## Average value of a function of three variables

Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.

## Average value of a function of three variables

If $f\left(x,y,z\right)$ is integrable over a solid bounded region $E$ with positive volume $V\left(E\right),$ then the average value of the function is

${f}_{\text{ave}}=\frac{1}{V\left(E\right)}\underset{E}{\iiint }f\left(x,y,z\right)dV.$

Note that the volume is $V\left(E\right)=\underset{E}{\iiint }1dV.$

## Finding an average temperature

The temperature at a point $\left(x,y,z\right)$ of a solid $E$ bounded by the coordinate planes and the plane $x+y+z=1$ is $T\left(x,y,z\right)=\left(xy+8z+20\right)\text{°}\text{C}\text{.}$ Find the average temperature over the solid.

Use the theorem given above and the triple integral to find the numerator and the denominator. Then do the division. Notice that the plane $x+y+z=1$ has intercepts $\left(1,0,0\right),\left(0,1,0\right),$ and $\left(0,0,1\right).$ The region $E$ looks like

$E=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 1-x,0\le z\le 1-x-y\right\}.$

Hence the triple integral of the temperature is

$\underset{E}{\iiint }f\left(x,y,z\right)dV=\underset{x=0}{\overset{x=1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{y=0}{\overset{y=1-x}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{z=0}{\overset{z=1-x-y}{\int }}\left(xy+8z+20\right)dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx=\frac{147}{40}.$

The volume evaluation is $V\left(E\right)=\underset{E}{\iiint }1dV=\underset{x=0}{\overset{x=1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{y=0}{\overset{y=1-x}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{z=0}{\overset{z=1-x-y}{\int }}1dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx=\frac{1}{6}.$

Hence the average value is ${T}_{\text{ave}}=\frac{147\text{/}40}{1\text{/}6}=\frac{6\left(147\right)}{40}=\frac{441}{20}$ degrees Celsius.

Find the average value of the function $f\left(x,y,z\right)=xyz$ over the cube with sides of length $4$ units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.

${f}_{\text{ave}}=8$

## Key concepts

• To compute a triple integral we use Fubini’s theorem, which states that 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
$\underset{B}{\iiint }f\left(x,y,z\right)dV=\underset{e}{\overset{f}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{c}{\overset{d}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{a}{\overset{b}{\int }}f\left(x,y,z\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz$

and is also equal to any of the other five possible orderings for the iterated triple integral.
• To compute the volume of a general solid bounded region $E$ we use the triple integral
$V\left(E\right)=\underset{E}{\iiint }1dV.$
• Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
• To compute the average value of a function over a general three-dimensional region, we use
${f}_{\text{ave}}=\frac{1}{V\left(E\right)}\underset{E}{\iiint }f\left(x,y,z\right)dV.$

## Key equations

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

In the following exercises, evaluate the triple integrals over the rectangular solid box $B.$

$\underset{B}{\iiint }\left(2x+3{y}^{2}+4{z}^{3}\right)dV,$ where $B=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 2,0\le z\le 3\right\}$

$192$

$\underset{B}{\iiint }\left(xy+yz+xz\right)dV,$ where $B=\left\{\left(x,y,z\right)|1\le x\le 2,0\le y\le 2,1\le z\le 3\right\}$

$\underset{B}{\iiint }\left(x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y+z\right)dV,$ where $B=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le \pi ,-1\le z\le 1\right\}$

$0$

$\underset{B}{\iiint }\left(z\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x+{y}^{2}\right)dV,$ where $B=\left\{\left(x,y,z\right)|0\le x\le \pi ,0\le y\le 1,-1\le z\le 2\right\}$

In the following exercises, change the order of integration by integrating first with respect to $z,$ then $x,$ then $y.$

$\underset{0}{\overset{1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{1}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{2}{\overset{3}{\int }}\left({x}^{2}+\text{ln}\phantom{\rule{0.2em}{0ex}}y+z\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz$

$\underset{1}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{2}{\overset{3}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{1}{\int }}\left({x}^{2}+\text{ln}\phantom{\rule{0.2em}{0ex}}y+z\right)dz\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy=\frac{35}{6}+2\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}2$

$\underset{0}{\overset{1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{-1}{\overset{1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{3}{\int }}\left(z{e}^{x}+2y\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz$

$\underset{-1}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{1}{\overset{3}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{4}{\int }}\left({x}^{2}z+\frac{1}{y}\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz$

$\underset{1}{\overset{3}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{4}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{-1}{\overset{2}{\int }}\left({x}^{2}z+\frac{1}{y}\right)dz\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy=64+12\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}3$

$\underset{1}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{-2}{\overset{-1}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{1}{\int }}\frac{x+y}{z}dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz$

Let $F,G,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}H$ be continuous functions on $\left[a,b\right],\left[c,d\right],$ and $\left[e,f\right],$ respectively, where $a,b,c,d,e,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}f$ are real numbers such that $a Show that

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

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
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
Got questions? Join the online conversation and get instant answers!