Chapter 15 – Multiple Integrals 15.7 Triple Integrals 1 Objectives: Understand how to calculate triple integrals Understand and apply the use of triple.

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Chapter 15 – Multiple Integrals15.7 Triple Integrals

15.7 Triple Integrals

Objectives: Understand how to

calculate triple integrals Understand and apply

the use of triple integrals to different applications

15.7 Triple Integrals 2

Triple Integrals Just as we defined single integrals for

functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.


Triple IntegralsLet’s first deal with the simplest case where f

is defined on a rectangular box:

, , , ,B x y z a x b c y d r z s


Triple IntegralsThe first step is

to divide B into sub-boxes—by dividing:

◦ The interval [a, b] into l subintervals [xi-1, xi] of equal width Δx.

◦ [c, d] into m subintervals of width Δy.◦ [r, s] into n subintervals of width Δz.


Triple IntegralsThe planes through

the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes

◦ Each sub-box has volume ΔV = Δx Δy Δz

1 1 1, , ,ijk i i j j k kB x x y y z z


Triple IntegralsThen, we form the triple Riemann sum

where the sample point is in Bijk.

* * *

1 1 1

, ,l m n

ijk ijk ijki j k

f x y z V

* * *, ,ijk ijk ijkx y z


Triple IntegralsThe triple integral of f over the box B is:

if this limit exists.◦ Again, the triple integral always exists if f

is continuous.

* * *

, ,1 1 1

, , lim , ,l m n

ijk ijk ijkl m n

i j kB

f x y z dV f x y z V


Fubini’s Theorem for Triple Integrals Just as for double integrals, the practical

method for evaluating triple integrals is to express them as iterated integrals, as follows.

If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then

, , , ,s d b

r c aB

f x y z dV f x y z dx dy dz


Fubini’s Theorem

The iterated integral on the right side of Fubini’s Theorem means that we integrate in the following order:

1. With respect to x (keeping y and z fixed)2. With respect to y (keeping z fixed)3. With respect to z

, , , ,s d b

r c aB

f x y z dV f x y z dx dy dz


Example 1 – pg. 998 # 4Evaluate the triple integral.

1 2

0 0

2yx

x

xyz dz dy dx


Integral over a Bounded Region

We restrict our attention to:

◦Continuous functions f

◦Certain simple types of regions


Type I Region Eq.5A solid region E is said to be of type 1 if it

lies between the graphs of two continuous functions of x and y. That is,

where D is the projection of E onto the xy-plane.

1 2, , , , , ,E x y z x y D u x y z u x y


Type I RegionNotice that:

◦ The upper boundary of the solid E is the surface with equation z = u2(x, y).

◦ The lower boundary is the surface z = u1(x, y).


Type I Region Eq. 6 If E is a type 1 region given by Equation 5,

then we have Equation 6:

2

1

,

,, , , ,

u x y

u x yE D

f x y z dV f x y z dz dA


Type II RegionA solid region E is said to be of type 1 if it

lies between the graphs of two continuous functions of x and y. That is,

where D is the projection of E onto the yz-plane.

1 2, , , , , ,E x y z y z D u y z x u y z


Type II RegionNotice that:

◦ The back surface is x = u1(y, z).

◦ The front surface is x = u2(y, z).


Type II Region Eq. 10For this type of region we have:

2

1

,

,, , , ,

u y z

u y zE D

f x y z dV f x y z dx dA


Type III RegionFinally, a type 3 region is of the form

where:◦ D is the projection of E

onto the xz-plane.

1 2, , , , ( , ) ,E x y z x z D u x z y u x z


Type III RegionNotice that:

◦ y = u1(x, z) is the left surface.

◦ y = u2(x, z) is the right surface.


Type III Region Eq. 11For this type of region, we have:

2

1

,

,, , , ,

u x z

u x zE D

f x y z dV f x y z dy dA


VisualizationRegions of Integration in Triple Int

egrals

http://www.cengage.com/math/discipline_content/stewartcalcet7/2008/14_cengage_tec/publish/deployments/transcendentals_7e/v15_7.html

http://www.cengage.com/math/discipline_content/stewartcalcet7/2008/14_cengage_tec/publish/deployments/transcendentals_7e/v15_7.html


Example 2 – pg. 998 # 10Evaluate the triple integral.

5cos , where

, , |0 1, 0 , 2

E

yz x dV

E x y z x y x x z x


Example 3 – pg. 998 # 20Use a triple integral to find the

volume of the given solid.

2

The solid bounded by the cylinder

and the planes 0, 4,

and 9.

y x z z

y


Example 4 – pg. 998 # 22Use a triple integral to find the

volume of the given solid.

2 2

The solid enclosed by the paraboloid

+ and the plane 16.x y z x


Example 5 – pg. 999 # 36Write five other iterated integrals

that are equal to the given iterated integral.

21

0 0 0

, ,yx

f x y z dz dy dx


More Examples

The video examples below are from section 15.7 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 3◦Example 6

http://www.wadsworthmedia.com/math/stewart/solution_videos/7et_15_07_01.html



Chapter 15 – Multiple Integrals 15.7 Triple Integrals 1 Objectives: Understand how to calculate triple integrals Understand and apply the use of triple.

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