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

Section 15.6 Triple Integrals 1031
Now we define the triple integral over a general bounded region E in threedimensional
space (a solid) by much the same procedure that we used for double integrals
(15.2.2). We enclose E in a box B of the type given by Equation 1. Then we define
F so that it agrees with f on E but is 0 for points in B that are outside E. By definition,
y y f sx, y, zd dV − y E
B
y Fsx, y, zd dV
z
E
z=u(x, y)
This integral exists if f is continuous and the boundary of E is “reasonably smooth.” The
triple integral has essentially the same properties as the double integral (Properties 6–9
in Section 15.2).
We restrict our attention to continuous functions f and to certain simple types of
regions. 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,
x
0
D
FIGURE 2
A type 1 solid region
z=u¡(x, y)
y
5 E − hsx, y, zd | sx, yd [ D, u 1sx, yd < z < u 2 sx, ydj
where D is the projection of E onto the xy-plane as shown in Figure 2. Notice that the
upper boundary of the solid E is the surface with equation z − u 2 sx, yd, while the lower
boundary is the surface z − u 1 sx, yd.
By the same sort of argument that led to (15.2.3), it can be shown that if E is a type 1
region given by Equation 5, then
b
x
a
z
0
y=g¡(x)
FIGURE 3
A type 1 solid region where the
projection D is a type I plane region
D
E
z=u(x, y)
z=u¡(x, y)
y=g(x)
y
6 y y f sx, y, zd dV − y Fy
E
D
u2sx, yd
u1sx, yd
f sx, y, zd dzG dA
The meaning of the inner integral on the right side of Equation 6 is that x and y are held
fixed, and therefore u 1 sx, yd and u 2 sx, yd are regarded as constants, while f sx, y, zd is
integrated with respect to z.
In particular, if the projection D of E onto the xy-plane is a type I plane region (as in
Figure 3), then
E − hsx, y, zd | a < x < b, t 1sxd < y < t 2 sxd, u 1 sx, yd < z < u 2 sx, ydj
and Equation 6 becomes
z
z=u(x, y)
7 y y f sx, y, zd dV − y b
E
a yt2sxd t1sxd
yd
yu2sx,
u1sx, yd
f sx, y, zd dz dy dx
x
0
E
c
D
x=h¡(y)
d
x=h(y)
z=u¡(x, y)
FIGURE 4
A type 1 solid region with a type II
projection
y
If, on the other hand, D is a type II plane region (as in Figure 4), then
E − hsx, y, zd | c < y < d, h 1syd < x < h 2 syd, u 1 sx, yd < z < u 2 sx, ydj
and Equation 6 becomes
8 y y f sx, y, zd dV − y d
c
E
yh2syd h1syd
yd
yu2sx,
u1sx, yd
f sx, y, zd dz dx dy
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