James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1030 Chapter 15 Multiple Integrals
3 Definition The triple integral of f over the box B is
if this limit exists.
y y f sx, y, zd dV − lim o m
o n
l, m, n l ` ol f sx i * jk , y i * jk , z i * jk d DV
i−1 j−1 k−1
B
Again, the triple integral always exists if f is continuous. We can choose the sample
point to be any point in the sub-box, but if we choose it to be the point sx i , y j , z k d we get
a simpler-looking expression for the triple integral:
y y f sx, y, zd dV − lim o m
o n
l, m, n l ` ol f sx i , y j , z k d DV
i−1 j−1 k−1
B
Just as for double integrals, the practical method for evaluating triple integrals is to
express them as iterated integrals as follows.
4 Fubini’s Theorem for Triple Integrals If f is continuous on the rectangular
box B − fa, bg 3 fc, dg 3 fr, sg, then
y y f sx, y, zd dV − y s
r yd c
yb f sx, y, zd dx dy dz
a
B
The iterated integral on the right side of Fubini’s Theorem means that we integrate
first with respect to x (keeping y and z fixed), then we integrate with respect to y (keeping
z fixed), and finally we integrate with respect to z. There are five other possible orders in
which we can integrate, all of which give the same value. For instance, if we integrate
with respect to y, then z, and then x, we have
y y f sx, y, zd dV − y b
a ys r yd f sx, y, zd dy dz dx
c
B
ExamplE 1 Evaluate the triple integral yyy B
xyz 2 dV, where B is the rectangular box
given by
B − hsx, y, zd | 0 < x < 1, 21 < y < 2, 0 < z < 3j
SOLUTION We could use any of the six possible orders of integration. If we choose to
integrate with respect to x, then y, and then z, we obtain
y y xyz 2 dV − y 3
3
0 y2 21 y1 xyz 2 dx dy dz −
0
B
y
0 21F x 2 yz 2 x−1
y2 dy dz
2
Gx−0
− y 3
0 y2 21
yz 2
2 dy dz − y 3
0
F
y 2 z 2 y−2
4
Gy−21
dz
− y 3
0
3z 2
4
dz −
z3
3
4G0
− 27 4
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