<< Chapter < Page Chapter >> Page >

For a , b , c , d , e , and f real numbers, the iterated triple integral can be expressed in six different orderings:

e f c d a b f ( x , y , z ) d x d y d z = e f ( c d ( a b f ( x , y , z ) d x ) d y ) d z = c d ( e f ( a b f ( x , y , z ) d x ) d z ) d y = a b ( e f ( c d f ( x , y , z ) d y ) d z ) d x = e f ( a b ( c d f ( x , y , z ) d y ) d x ) d z = c e ( a b ( e f f ( x , y , z ) d z ) d x ) d y = a b ( c e ( e f f ( x , y , z ) d z ) d y ) d x .

For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini’s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).

Evaluating a triple integral

Evaluate the triple integral z = 0 z = 1 y = 2 y = 4 x = −1 x = 5 ( x + y z 2 ) d x d y d z .

The order of integration is specified in the problem, so integrate with respect to x first, then y , and then z .

z = 0 z = 1 y = 2 y = 4 x = −1 x = 5 ( x + y z 2 ) d x d y d z = z = 0 z = 1 y = 2 y = 4 [ x 2 2 + x y z 2 | x = −1 x = 5 ] d y d z Integrate with respect to x . = z = 0 z = 1 y = 2 y = 4 [ 12 + 6 y z 2 ] d y d z Evaluate. = z = 0 z = 1 [ 12 y + 6 y 2 2 z 2 | y = 2 y = 4 ] d z Integrate with respect to y . = z = 0 z = 1 [ 24 + 36 z 2 ] d z Evaluate. = [ 24 z + 36 z 3 3 ] z = 0 z = 1 = 36 . Integrate with respect to z .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluating a triple integral

Evaluate the triple integral B x 2 y z d V where B = { ( x , y , z ) | 2 x 1 , 0 y 3 , 1 z 5 } as shown in the following figure.

In x y z space, there is a box given with corners (1, 0, 5), (1, 0, 1), (1, 3, 1), (1, 3, 5), (negative 2, 0, 5), (negative 2, 0, 1), (negative 2, 3, 1), and (negative 2, 3, 5).
Evaluating a triple integral over a given rectangular box.

The order is not specified, but we can use the iterated integral in any order without changing the level of difficulty. Choose, say, to integrate y first, then x , and then z .

B x 2 y z d V = 1 5 −2 1 0 3 [ x 2 y z ] d y d x d z = 1 5 −2 1 [ x 2 y 2 2 z | 0 3 ] d x d z = 1 5 −2 1 9 2 x 2 z d x d z = 1 5 [ 9 2 x 3 3 z | −2 1 ] d z = 1 5 27 2 z d z = 27 2 z 2 2 | 1 5 = 162.

Now try to integrate in a different order just to see that we get the same answer. Choose to integrate with respect to x first, then z , and then y .

B x 2 y z d V = 0 3 1 5 −2 1 [ x 2 y z ] d x d z d y = 0 3 1 5 [ x 3 3 y z | −2 1 ] d z d y = 0 3 1 5 3 y z d z d y = 0 3 [ 3 y z 2 2 | 1 5 ] d y = 0 3 36 y d y = 36 y 2 2 | 0 3 = 18 ( 9 0 ) = 162.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate the triple integral B z sin x cos y d V where B = { ( x , y , z ) | 0 x π , 3 π 2 y 2 π , 1 z 3 } .

B z sin x cos y d V = 8

Got questions? Get instant answers now!

Triple integrals over a general bounded region

We now expand the definition of the triple integral to compute a triple integral over a more general bounded region E in 3 . The general bounded regions we will consider are of three types. First, let D be the bounded region that is a projection of E onto the x y -plane. Suppose the region E in 3 has the form

E = { ( x , y , z ) | ( x , y ) D , u 1 ( x , y ) z u 2 ( x , y ) } .

For two functions z = u 1 ( x , y ) and z = u 2 ( x , y ) , such that u 1 ( x , y ) u 2 ( x , y ) for all ( x , y ) in D as shown in the following figure.

In x y z space, there is a shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the x y plane as region D.
We can describe region E as the space between u 1 ( x , y ) and u 2 ( x , y ) above the projection D of E onto the x y -plane.

Triple integral over a general region

The triple integral of a continuous function f ( x , y , z ) over a general three-dimensional region

E = { ( x , y , z ) | ( x , y ) D , u 1 ( x , y ) z u 2 ( x , y ) }

in 3 , where D is the projection of E onto the x y -plane, is

E f ( x , y , z ) d V = D [ u 1 ( x , y ) u 2 ( x , y ) f ( x , y , z ) d z ] d A .

Similarly, we can consider a general bounded region D in the x y -plane and two functions y = u 1 ( x , z ) and y = u 2 ( x , z ) such that u 1 ( x , z ) u 2 ( x , z ) for all ( x , z ) in D . Then we can describe the solid region E in 3 as

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask