Section 16.7

Triple Integrals

TRIPLE INTEGRAL OVER A BOX

Consider a function w = f (x, y, z) of three variables defined on the box B given by

Divide B into sub-boxes by dividing the interval [a, b] into l subintervals of equal width Δx, dividing [c, d] into m subintervals of equal width Δy, and dividing [r, s] into n subintervals of equal width Δz. This divides the box B into l∙m∙n sub-boxes. A typical sub-box Bilk is

Bijk = [xi − 1, xi] × [yj − 1, yj] × [zk − 1, zk]

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

],[],[],[

},,|),,{(

srdcba

szrdycbxazyxB

We now form the triple Riemann sum

where the sample point is in box Bijk.

TRIPLE INTEGRAL OVER A BOX (CONTINUED)

l

i

m

j

n

kijkijkijk Vzyxf

1 1 1

*** ),,(

),,( ***ijkijkijk zyx

The triple integral of f over the box B is

if this limit exists.

The triple integral always exists if f is continuous. If we choose the sample point to be (xi, yj, zk), the triple integral simplifies to

TRIPLE INTEGRAL OVER A BOX (CONCLUDED)

l

i

m

j

n

kijkijkijk

Bnml

VzyxfdVzyxf1 1 1

***

,,),,(lim),,(

l

i

m

j

n

kkji

Bnml

VzyxfdVzyxf1 1 1

,,),,(lim),,(

FUBINI’S THEOREM FOR TRIPLE INTEGRAL

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

s

r

d

c

b

aB

dzdydxzyxfdVzyxf ),,(),,(

NOTE: The order of the partial antiderivatives does not matter as long as the endpoints correspond to the proper variable.

EXAMPLE

Evaluate the triple integral , where

B is the rectangular box given by

B = {(x, y, z) | 1 ≤ x ≤ 2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 2}

B

dVzyx2

TRIPLE INTEGRAL OVER A BOUNDED REGION

The triple integral over the bounded region E is defined as

where B is a box containing the region E and the function F is defined as

BE

dVzyxFdVzyxf ),,(),,(

EBzyx

EzyxzyxfzyxF

innotbutis),,(if0

inis),,(if),,(),,(

The region E is said to by of type 1 if it lies between to continuous functions of x and y. That is,

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

The triple integral over a type 1 region is

TYPE 1 REGIONS

D

yxu

yxuE

dAdzzyxfdVzyxf).(

),(

2

1

),,(),,(

)},(),(,),(|),,{( 21 yxuzyxuDyxzyxE

If D is a type I region in the xy-plane, then E can be described as

and the triple integral becomes

TYPE 1 REGIONS (CONTINUED)

)},(),(),()(,|),,{( 2121 yxuzyxuxgyxgbxazyxE

E

b

a

xg

xg

yxu

yxudxdydzzyxfdVzyxf

)(

)(

),.(

),(

2

1

2

1

),,(),,(

If D is a type II region in the xy-plane, then E can be described as

and the triple integral becomes

TYPE 1 REGIONS (CONCLUDED)

)},(),(),()(,|),,{( 2121 yxuzyxuyhxyhdyczyxE

E

d

c

yh

yh

yxu

yxudydxdzzyxfdVzyxf

)(

)(

),.(

),(

2

1

2

1

),,(),,(

EXAMPLE

Evaluate the triple integral , where

E is the region bounded by the planes x = 0, y = 0, z = 0, and 2x + 2y + z = 4.

E

dVy

The region E is said to by of type 2 if it lies between two continuous functions of y and z. That is,

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

The triple integral over a type 2 region is

TYPE 2 REGIONS

D

zyu

zyuE

dAdxzyxfdVzyxf),(

),(

2

1

),,(),,(

)},(),(,),(|),,{( 21 zyuxzyuDzyzyxE

The region E is said to by of type 3 if it lies between two continuous functions of x and z. That is,

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

The triple integral over a type 3 region is

TYPE 3 REGIONS

D

zxu

zxuE

dAdyzyxfdVzyxf).(

),(

2

1

),,(),,(

)},(),(,),(|),,{( 21 zxuyzxuDzxzyxE

EXAMPLE

Evaluate the triple integral ,

where E is the region bounded by the paraboloid x = y2 + z2 and the plane x = 4.

E

dVzy 22

VOLUME AND TRIPLE INTEGRALS

The triple integral of the function f (x, y, z) = 1 over the region E gives the volume of E; that is,

E

dVEV 1)(

EXAMPLE

Find the volume of the region E bounded by the plane z = 0, the plane z = x, and the cylinder x = 4 − y2.

MASS

Suppose the density function of a solid object that occupies the region E is ρ(x, y, z). Then the mass of the solid is

E

dVzyxm ),,(

MOMENTS

Suppose the density function of a solid object that occupies the region E is ρ(x, y, z). Then the moments of the solid about the three coordinate planes are

E

xy

E

xz

E

yz

dVzyxzM

dVzyxyMdVzyxxM

),,(

),,(),,(

CENTER OF MASS

),,( zyxThe center of mass is located at the point

where

m

Mz

m

My

m

Mx xyxzyz

If the density is constant, the center of mass of the solid is called the centroid of E.

MOMENTS OF INERTIA

The moments of inertia about the three coordinate axis are

E

z

E

y

E

x

dVzyxyxI

dVzyxzxI

dVzyxzyI

),,()(

),,()(

),,()(

22

22

22

EXAMPLE

Find the mass and center of mass of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1 whose density function is given by ρ(x, y, z) = y.