13.8 Triple Integrals
In this section, we dene triple integrals for functions of three variables. Let f (x, y, z)
is dened on a rectangular box
B = [a, b] × [c, d] × [r, s] = { (x, y, z) | a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s }.
The rst step is to divide B into sub-boxes. We
accomplish this by dividing the intervals [a, b], [c, d]
and [r, s] as follows:
a =x0 < x1 < ... < xl−1 < xl = b
c =y0 < y1 < ... < ym−1 < ym = d
r =z0 < z1 < ... < zn−1 < zn = s
The planes through these partition points parallel
to coordinate planes divide the box B into l · m · n
sub-boxes
Bijk = [xi−1 , xi ] × [yj−1 , yj ] × [zk−1 , zk ]
with volume ∆Vijk = ∆xi ∆yj ∆zk ,
where ∆xi = xi −xi−1 , ∆yj = yj −yj−1 and ∆zk =
zk − zk−1 .
∗
∗
Next we choose a sample point (x∗ijk , yijk
, zijk
)
in each Bijk and form the triple Riemann sum
n
m X
l X
X
.
∗
∗
f (x∗ijk , yijk
, zijk
)∆V
i=1 j=1 k=1
Let the norm kP k of the partition P to be the length of the longest diagonal of all
the boxes Bijk .
Denition. The triple integral of f over the box B is
˚
f (x, y, z) dV := lim
kP k→0
B
if this limit exists.
l X
m X
n
X
i=1 j=1 k=1
∗
∗
f (x∗ijk , yijk
, zijk
)∆Vijk
Fubini's Theorem for triple integrals. If f is continuous on the rectangular box
B = [a, b] × [c, d] × [r, s], then
˚
ˆ sˆ
f (x, y, z) dV =
r
B
c
dˆ b
f (x, y, z) dxdydz
a
and there are 5 other possible orders in which we can integrate.
Example 1. Let B = { (x, y, z) | 0 ≤ x ≤ 2, −3 ≤ y ≤ 0, −1 ≤ z ≤ 1 }. Evaluate
the integral
˚
(x2 + yz) dV
B
Now we dene the triple integral over a general bounded region E in threedimensional 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
˚
¨ "ˆ
f (x, y, z) dV =
E
#
φ2 (x,y)
f (x, y, z) dz dA
D
φ1 (x,y)
A solid region E is of type 2 if it is of the form,
E = { (x, y, z) | (y, z) ∈ D, φ1 (y, z) ≤ x ≤ φ2 (y, z) }
where D is the projection of E onto the yz -plane.
Then
˚
¨ "ˆ
f (x, y, z) dV =
E
#
φ2 (y,z)
f (x, y, z) dx dA
D
φ1 (y,z)
A solid region E is of type 3 if it is of the form,
E = { (x, y, z) | (x, z) ∈ D, φ1 (x, z) ≤ y ≤ φ2 (x, z) }
where D is the projection of E onto the xz -plane.
Then
˚
¨ "ˆ
φ2 (x,z)
f (x, y, z) dy dA
f (x, y, z) dV =
E
#
D
φ1 (x,z)
When we set up a triple integral it is wise to draw two diagrams: one of the solid
region E and one of its projection D on the corresponding coordinate plane.
Example 2. Evaluate the integral
y = 0, z = 0 and 3x + 2y + z = 6.
˝
E
x dV where E is bounded the planes x = 0,
Fact: The volume of the solid E is given by the integral,
˚
dV
E
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.
Fact: The mass of the solid E with variable density ρ(x, y, z) is given by the integral
˚
m=
ρ(x, y, z) dV
E
The center of mass is located at the point (x̄, ȳ, z̄), where
˝
E
x̄ = ˝
E
xρ(x, y, z) dV
ρ(x, y, z) dV
˝
E
ȳ = ˝
E
yρ(x, y, z) dV
ρ(x, y, z) dV
˝
E
z̄ = ˝
zρ(x, y, z) dV
ρ(x, y, z) dV
E
Example 4. Find the mass and the center of masspof the solid bounded by x = y 2 + z 2
and x = 4 if the density function is ρ(x, y, z) = y 2 + z 2 .