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

1032 Chapter 15 Multiple Integrals
z
(0, 0, 1)
z=1-x-y
E
0
(0, 1, 0)
(1, 0, 0)
y
z=0
x
FIGURE 5
y
1
y=1-x
D
0 y=0 1 x
FIGURE 6
z
ExamplE 2 Evaluate yyy E
z dV, where E is the solid tetrahedron bounded by the four
planes x − 0, y − 0, z − 0, and x 1 y 1 z − 1.
SOLUTION When we set up a triple integral it’s wise to draw two diagrams: one of
the solid region E (see Figure 5) and one of its projection D onto the xy-plane (see
Fig ure 6). The lower boundary of the tetrahedron is the plane z − 0 and the upper
boundary is the plane x 1 y 1 z − 1 (or z − 1 2 x 2 y), so we use u 1 sx, yd − 0 and
u 2 sx, yd − 1 2 x 2 y in Formula 7. Notice that the planes x 1 y 1 z − 1 and z − 0
intersect in the line x 1 y − 1 (or y − 1 2 x) in the xy-plane. So the projection of E is
the triangular region shown in Figure 6, and we have
9
E − hsx, y, zd | 0 < x < 1, 0 < y < 1 2 x, 0 < z < 1 2 x 2 yj
This description of E as a type 1 region enables us to evaluate the integral as follows:
z−12x2y
z2
dy dx
2Gz−0
y y z dV − y 1
y 12x2y
1
0 y12x z dz dy dx −
0 0
E
y F
0 y12x
0
− 1 2 y 1
− 1 6 y 1
0 y12x 0
0
s1 2 x 2 yd 2 dy dx − 1 2 y 1
0
F2
s1 2 xd 3 dx − 1 1
s1 2 xd4
F2 − 1 6 4 24
G0
A solid region E is of type 2 if it is of the form
y−12x
s1 2 x 2 yd3
dx
3
Gy−0
■
x
0
E
FIGURE 7
A type 2 region
x=u(y, z)
D
x=u¡(y, z)
y
E − hsx, y, zd | sy, zd [ D, u 1sy, zd < x < u 2 sy, zdj
where, this time, D is the projection of E onto the yz-plane (see Figure 7). The back surface
is x − u 1 sy, zd, the front surface is x − u 2 sy, zd, and we have
10 y y f sx, y, zd dV − y Fy
E
D
Finally, a type 3 region is of the form
u2sy, zd
u1sy, zd
f sx, y, zd dxG dA
E − hsx, y, zd | sx, zd [ D, u 1sx, zd < y < u 2 sx, zdj
z
y=u(x, z)
where D is the projection of E onto the xz-plane, y − u 1 sx, zd is the left surface, and
y − u 2 sx, zd is the right surface (see Figure 8). For this type of region we have
D
E
11 y y f sx, y, zd dV − y Fy
E
D
u2sx, zd
u1sx, zd
f sx, y, zd dyG dA
x
0
y=u¡(x, z)
y
In each of Equations 10 and 11 there may be two possible expressions for the integral
depending on whether D is a type I or type II plane region (and corresponding to Equations
7 and 8).
FIGURE 8
A type 3 region
ExamplE 3 Evaluate yyy E sx 2 1 z 2 dV, where E is the region bounded by the paraboloid
y − x 2 1 z 2 and the plane y − 4.
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