c Dr Oksana Shatalov, Spring 2014
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13.8: Triple Integrals
Mass problem: Given a solid object, that occupies the region B in R3 , with
density ρ(x, y, z). Find the mass of the object.
Solution: Let B be a rectangular box:
B = {(x, y, z)|a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s} .
Partition in sub-boxes:
mijk = ρ(x∗i , yj∗ , zk∗ )∆Vijk
q
kP k = max ∆x2i + ∆yj2 + ∆zk2
XXX
m = lim
ρ(x∗i , yj∗ , zk∗ )∆Vijk
kP k→0
i
j
k
ZZZ
m=
ρ(x, y, z) dV
1
B
FUBINI’s THEOREM: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) dx dydz
B
r
c
a
and there are 5 other possible orders in which we can integrate.
EXAMPLE 1. Let B = [0, 1] × [−1, 3] × [0, 3]. Evaluate
ZZZ
I=
xyeyz dV
B
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All figures are from the course textbook
c Dr Oksana Shatalov, Spring 2014
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FACT: The volume of the solid E is given by the integral,
ZZZ
V =
dV.
E
FACT: The mass of the solid E with variable density ρ(x, y, z) is given by the
integral,
m=
.
EXAMPLE 2. Find the mass of the solid p
bounded by x = y 2 + z 2 and the plane
x = 4 if the density function is ρ(x, y, z) = y 2 + z 2 .
c Dr Oksana Shatalov, Spring 2014
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EXAMPLE 3. Use a triple integral to find the volume of the solid bounded by
the surfaces z = x2 + y 2 and z = 5 − 4x2 − 4y 2 .
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EXAMPLE 4. Use a triple integral to find the volume of the solid bounded by
the elliptic cylinder 4x2 + z 2 = 4 and the planes y = 0 and y = z + 2.
E = {(x, y, z)|(y, z) ∈ D, φ1 (y, z) ≤ x ≤ φ2 (y, z)}
E = {(x, y, z)|(x, y) ∈ D, φ1 (x, y) ≤ z ≤ φ2 (x, y)}
E
f (x, y, z) dV =
D
#
f (x, y, z) dz dA
φ2 (x,y)
φ1 (x,y)
Z Z "Z
ZZZ
E
f (x, y, z) dV =
Z Z Z
f (x, y, z) dx dA
ZZZ
E
f (x, y, z) dV =
where D is the projection of E onto the xzplane.
E = {(x, y, z)|(x, z) ∈ D, φ1 (x, z) ≤ y ≤ φ2 (x, z)}
A solid region of TYPE III:
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 on the corresponding
coordinate plane.
ZZZ
where D is the projection of E onto the xy- where D is the projection of E onto the yzplane.
plane.
A solid region of TYPE II:
A solid region of TYPE I:
Table 1: Triple integrals over a general bounded region E
c Dr Oksana Shatalov, Spring 2014
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