<< Chapter < Page Chapter >> Page >

Changing the order of integration

As we have already seen in double integrals over general bounded regions, changing the order of the integration is done quite often to simplify the computation. With a triple integral over a rectangular box, the order of integration does not change the level of difficulty of the calculation. However, with a triple integral over a general bounded region, choosing an appropriate order of integration can simplify the computation quite a bit. Sometimes making the change to polar coordinates can also be very helpful. We demonstrate two examples here.

Changing the order of integration

Consider the iterated integral

x = 0 x = 1 y = 0 y = x 2 z = 0 z = y f ( x , y , z ) d z d y d x .

The order of integration here is first with respect to z , then y , and then x . Express this integral by changing the order of integration to be first with respect to x , then z , and then y . Verify that the value of the integral is the same if we let f ( x , y , z ) = x y z .

The best way to do this is to sketch the region E and its projections onto each of the three coordinate planes. Thus, let

E = { ( x , y , z ) | 0 x 1 , 0 y x 2 , 0 z y } .

and

x = 0 x = 1 y = 0 y = x 2 z = 0 z = y 2 f ( x , y , z ) d z d y d x = E f ( x , y , z ) d V .

We need to express this triple integral as

y = c y = d z = v 1 ( y ) z = v 2 ( y ) x = u 1 ( y , z ) x = u 2 ( y , z ) f ( x , y , z ) d x d z d y .

Knowing the region E we can draw the following projections ( [link] ):

on the x y -plane is D 1 = { ( x , y ) | 0 x 1 , 0 y x 2 } = { ( x , y ) | 0 y 1 , y x 1 } ,

on the y z -plane is D 2 = { ( y , z ) | 0 y 1 , 0 z y 2 } , and

on the x z -plane is D 3 = { ( x , z ) | 0 x 1 , 0 z x 2 } .

Three similar versions of the following graph are shown: In the x y plane, a region D1 is bounded by the x axis, the line x = 1, and the curve y = x squared. In the second version, region D2 on the z y plane is shown with equation z = y squared. And in the third version, region D3 on the x z plane is shown with equation z = x squared.
The three cross sections of E on the three coordinate planes.

Now we can describe the same region E as { ( x , y , z ) | 0 y 1 , 0 z y 2 , y x 1 } , and consequently, the triple integral becomes

y = c y = d z = v 1 ( y ) z = v 2 ( y ) x = u 1 ( y , z ) x = u 2 ( y , z ) f ( x , y , z ) d x d z d y = y = 0 y = 1 z = 0 z = x 2 x = y x = 1 f ( x , y , z ) d x d z d y .

Now assume that f ( x , y , z ) = x y z in each of the integrals. Then we have

x = 0 x = 1 y = 0 y = x 2 z = 0 z = y 2 x y z d z d y d x = x = 0 x = 1 y = 0 y = x 2 [ x y z 2 2 | z = 0 z = y 2 ] d y d x = x = 0 x = 1 y = 0 y = x 2 ( x y 5 2 ) d y d x = x = 0 x = 1 [ x y 6 12 | y = 0 y = x 2 ] d x = x = 0 x = 1 x 13 12 d x = 1 168 , y = 0 y = 1 z = 0 z = y 2 x = y x = 1 x y z d x d z d y = y = 0 y = 1 z = 0 z = y 2 [ y z x 2 2 | y 1 ] d z d y = y = 0 y = 1 z = 0 z = y 2 ( y z 2 y 2 z 2 ) d z d y = y = 0 y = 1 [ y z 2 4 y 2 z 2 4 | z = 0 z = y 2 ] d y = y = 0 y = 1 ( y 5 4 y 6 4 ) d y = 1 168 .

The answers match.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write five different iterated integrals equal to the given integral

z = 0 z = 4 y = 0 y = 4 z x = 0 x = y f ( x , y , z ) d x d y d z .

(i) z = 0 z = 4 x = 0 x = 4 z y = x 2 y = 4 z f ( x , y , z ) d y d x d z , (ii) y = 0 y = 4 z = 0 z = 4 y x = 0 x = y f ( x , y , z ) d x d z d y , (iii) y = 0 y = 4 x = 0 x = y z = 0 z = 4 y f ( x , y , z ) d z d x d y , (iv) x = 0 x = 2 y = x 2 y = 4 z = 0 z = 4 y f ( x , y , z ) d z d y d x , (v) x = 0 x = 2 z = 0 z = 4 x 2 y = x 2 y = 4 z f ( x , y , z ) d y d z d x

Got questions? Get instant answers now!

Changing integration order and coordinate systems

Evaluate the triple integral E x 2 + z 2 d V , where E is the region bounded by the paraboloid y = x 2 + z 2 ( [link] ) and the plane y = 4 .

The paraboloid y = x squared + z squared is shown opening up along the y axis to y = 4.
Integrating a triple integral over a paraboloid.

The projection of the solid region E onto the x y -plane is the region bounded above by y = 4 and below by the parabola y = x 2 as shown.

In the x y plane, the graph of y = x squared is shown with the line y = 4 intersecting the graph at (negative 2, 4) and (2, 4).
Cross section in the x y -plane of the paraboloid in [link] .

Thus, we have

E = { ( x , y , z ) | 2 x 2 , x 2 y 4 , y x 2 z y x 2 } .

The triple integral becomes

E x 2 + z 2 d V = x = −2 x = 2 y = x 2 y = 4 z = y x 2 z = y x 2 x 2 + z 2 d z d y d x .

This expression is difficult to compute, so consider the projection of E onto the x z -plane. This is a circular disc x 2 + z 2 4 . So we obtain

E x 2 + z 2 d V = x = −2 x = 2 y = x 2 y = 4 z = y x 2 z = y x 2 x 2 + z 2 d z d y d x = x = −2 x = 2 z = 4 x 2 z = 4 x 2 y = x 2 + z 2 y = 4 x 2 + z 2 d y d z d x .

Here the order of integration changes from being first with respect to z , then y , and then x to being first with respect to y , then to z , and then to x . It will soon be clear how this change can be beneficial for computation. We have

x = −2 x = 2 z = 4 x 2 z = 4 x 2 y = x 2 + z 2 y = 4 x 2 + z 2 d y d z d x = x = −2 x = 2 z = 4 x 2 z = 4 x 2 ( 4 x 2 z 2 ) x 2 + z 2 d z d x .

Now use the polar substitution x = r cos θ , z = r sin θ , and d z d x = r d r d θ in the x z -plane. This is essentially the same thing as when we used polar coordinates in the x y -plane, except we are replacing y by z . Consequently the limits of integration change and we have, by using r 2 = x 2 + z 2 ,

x = −2 x = 2 z = 4 x 2 z = 4 x 2 ( 4 x 2 z 2 ) x 2 + z 2 d z d x = θ = 0 θ = 2 π r = 0 r = 2 ( 4 r 2 ) r r d r d θ = 0 2 π [ 4 r 3 3 r 5 5 | 0 2 ] d θ = 0 2 π 64 15 d θ = 128 π 15 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
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
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com 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