<< Chapter < Page Chapter >> Page >
  • 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 ( x , y ) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f ( x , y , z ) of three variables over a rectangular solid box in space, 3 . Later in this section we extend the definition to more general regions in 3 .

Integrable functions of three variables

We can define a rectangular box B in 3 as B = { ( x , y , z ) | a x b , c y d , e z f } . We follow a similar procedure to what we did in Double Integrals over Rectangular Regions . We divide the interval [ a , b ] into l subintervals [ x i 1 , x i ] of equal length Δ x = x i x i 1 l , divide the interval [ c , d ] into m subintervals [ y i 1 , y i ] of equal length Δ y = y j y j 1 m , and divide the interval [ e , f ] into n subintervals [ z i 1 , z i ] of equal length Δ z = z k z k 1 n . Then the rectangular box B is subdivided into l m n subboxes B i j k = [ x i 1 , x i ] × [ y i 1 , y i ] × [ z i 1 , z i ] , as shown in [link] .

In x y z space, there is a box B with a subbox Bijk with sides of length Delta x, Delta y, and Delta z.
A rectangular box in 3 divided into subboxes by planes parallel to the coordinate planes.

For each i , j , and k , consider a sample point ( x i j k * , y i j k * , z i j k * ) in each sub-box B i j k . We see that its volume is Δ V = Δ x Δ y Δ z . Form the triple Riemann sum

i = 1 l j = 1 m k = 1 n f ( x i j k * , y i j k * , z i j k * ) Δ x Δ y 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.


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

lim l , m , n i = 1 l j = 1 m k = 1 n f ( x i j k * , y i j k * , z i j k * ) Δ x Δ y Δ z = B f ( x , y , z ) d V

if this limit exists.

When the triple integral exists on B , the function f ( x , y , z ) is said to be integrable on B . Also, the triple integral exists if f ( x , y , z ) 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 ( x i j k * , y i j k * , z i j k * ) can be any point in the rectangular sub-box B i j k 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 ( x , y , z ) is continuous on a rectangular box B = [ a , b ] × [ c , d ] × [ e , f ] , then

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

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

Questions & Answers

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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?