Chapter 13. Multiple integrals.
Section 13.8 Triple integrals.
We want to define the triple integrals for functions of three variables.
Let f is defined on a rectangular box
B = {(x, y, z)|a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s} = [a, b] × [c, d] × [r, s]
We partition the intervals [a, b], [c, d], and [r, s] as follows:
a = x0 < x1 < ... < xm = l
c = y0 < y1 < ... < yn = m
r = z0 < z1 < ... < zk = n
The planes through these partition points parallel to coordinate planes divide the box B into lmn
sub-boxes
Bijk = [xi−1 , xi ] × [yj−1 , yj ] × [zk−1 , zk ]
The volume of Bijk is
∆Vijk = ∆xi ∆yj ∆zk
where ∆xi = xi − xi−1 , ∆yj = yj − yj−1 , and ∆zk = zk − zk−1 .
Then we form the triple Riemann sum
l X
m X
n
X
∗
∗
f (x∗ijk , yijk
, zijk
)∆Vijk
i=1 j=1 k=1
∗
∗
where (x∗ijk , yijk
, zijk
) ∈ Bijk . We define the norm kP k of the partition P to be the length of the
longest diagonal of all the boxes Bijk .
Definition. The triple integral of f over the box B is
ZZZ
f (x, y, z)dV = lim
kP k→0
B
l X
m X
n
X
∗
∗
, zijk
)∆Vijk
f (x∗ijk , yijk
i=1 j=1 k=1
if this limit exists.
Fubini’s Theorem for triple integrals. If f is continuous on the rectangular box B = [a, b] ×
[c, d] × [r, s], then
ZZZ
Z sZ dZ b
f (x, y, z)dV =
f (x, y, z)dxdydz
B
r
c
1
a
There are five other possible orders
we can integrate.
RRRin which
2
Example 1. Evaluate the integral
(x + yz)dV , where
E
E = {(x, y, z)|0 ≤ x ≤ 2, −3 ≤ y ≤ 0, −1 ≤ z ≤ 1}
Now we define the triple integral over a general bounded region E in three-dimensional
space.
A 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,
E = {(x, y, z)|(x, y) ∈ D, ϕ1 (x, y) ≤ z ≤ ϕ2 (x, y)}
where D is the projection of E onto the xy-plane.
Then
Z Z "Z
ZZZ
ϕ2 (x,y)
f (x, y, z)dV =
E
#
f (x, y, z)dz dA
ϕ1 (x,y)
D
2
Example 2.
Evaluate
RRR
xdV, where E is bounded by planes x = 0, y = 0, z = 0, and
E
3x + 2y + z = 6.
3
A solid region E is of type 2 if it is of the form
E = {(x, y, z)|(y, z) ∈ D, ψ1 (y, z) ≤ x ≤ ψ2 (y, z)}
where D is the projection of E onto the yz-plane.
Then
Z Z "Z
ZZZ
ψ2 (y,z)
f (x, y, z)dV =
E
#
f (x, y, z)dx dA
ψ1 (y,z)
D
A solid region E is of type 3 if it is of the form
E = {(x, y, z)|(x, z) ∈ D, χ1 (x, z) ≤ y ≤ χ2 (x, z)}
where D is the projection of E onto the xz-plane.
Then
Z Z "Z
ZZZ
χ2 (x,z)
f (x, y, z)dV =
E
#
f (x, y, z)dy dA
χ1 (x,z)
D
Applications of triple integrals.
ZZZ
dV
V (E) =
E
4
Example 3. Find the volume of the solid bounded by the elliptic cylinder 4x2 + z 2 = 4 and
the planes y = 0 and y = z + 2.
5
If the density function of a solid object that occupies the region E is ρ(x, y, z) in units of mass
per unit volume, at any given point (x, y, z), then its mass is
ZZZ
ρ(x, y, z)dV
m=
E
and its moments about the three coordinate planes are
ZZZ
ZZZ
Myz =
xρ(x, y, z)dV,
Mxz =
yρ(x, y, z)dV,
E
E
Mxy =
zρ(x, y, z)dV
E
The center of mass is located at the point (x̄, ȳ, z̄) where
RRR
RRR
xρ(x, y, z)dV
yρ(x, y, z)dV
Myz
Mxz
E
E
x̄ =
= RRR
,
ȳ =
= RRR
,
m
m
ρ(x, y, z)dV
ρ(x, y, z)dV
E
ZZZ
E
RRR
zρ(x, y, z)dV
Mxy
E
z̄ =
= RRR
m
ρ(x, y, z)dV
E
Example 4. Find the center of mass of a solid E with density ρ(x, y, z) = x, where E is bounded
by the paraboloid x = 4y 2 + 4z 2 and the plane x = 4.
6