Solução_Calculo_Stewart_6e

F.
TX.10
SECTION 16.6 TRIPLE INTEGRALS ET SECTION 15.6 ¤ 247
47. (a) m = 1
0
√ 1−x 2
0
y
0
(b) (x, y, z) = m −1 1 √ 1−x 2
0 0
(c) I z =
1
0
3π
(1 + x + y + z) dz dy dx = + 11
32 24
m −1 1
0
y
0
√ 1−x 2
0
x(1 + x + y + z) dz dy dx,
y
0
y(1 + x + y + z) dz dy dx,
m −1 1 √
1−x 2 y z(1 + x + y + z) dz dy dx 0 0 0
28 30π +128 45π +208
= , ,
9π +44 45π +220 135π + 660
√ 1−x 2
0
y
0
(x 2 + y 2 )(1 + x + y + z) dz dy dx =
68 + 15π
240
49. (a) f(x, y, z) is a joint density function, so we know R 3 f(x, y, z) dV =1.Herewehave
R 3 f(x, y, z) dV = ∞
−∞
∞
−∞
Then we must have 8C =1 ⇒ C = 1 8 .
(b) P (X ≤ 1,Y ≤ 1,Z ≤ 1) = 1
−∞
1
−∞
= 1 8
∞
−∞ f(x, y, z) dz dy dx = 2
= C 2
0 xdx 2
0 ydy 2
0 zdz= C 1
2 x2 2
0
1 f(x, y, z) dz dy dx = 1
1
1
−∞ 0 0 0
1
0 xdx 1
0 ydy 1
0 zdz= 1 8
1
2 x2 1
0
1
2 y2 1
0
2
2
0 0 0
1
2 y2 2
0
1
xyz dz dy dx
8
Cxyz dz dy dx
1 z2 2
=8C
2 0
1
2 z2 1
0 = 1 8
1
2
3
= 1 64
(c) P (X + Y + Z ≤ 1) = P ((X, Y, Z) ∈ E) where E is the solid region in the first octant bounded by the coordinate planes
and the plane x + y + z =1. The plane x + y + z =1meets the xy-plane in the line x + y =1,sowehave
1−x
1−x−y
P (X + Y + Z ≤ 1) = f(x, y, z) dV = 1
1
E 0 0 0
= 1 1
1−x
8
xy 1
z2 z=1−x−y
0 0 2
dy dx = 1 z=0 16
51. V (E) =L 3 ⇒ f ave = 1 L 3 L
0
= 1 1
1−x
16 0
= 1 1
16 0
8 xyzdzdydx
1
0
1−x
0
xy(1 − x − y) 2 dy dx
[(x 3 − 2x 2 + x)y +(2x 2 − 2x)y 2 + xy 3 ] dy dx
0
(x 3 − 2x 2 + x) 1 2 y2 +(2x 2 − 2x) 1 3 y3 + x 1
y4 y=1−x
dx
4 y=0
= 1 1 (x −
192 0 4x2 +6x 3 − 4x 4 + x 5 ) dx = 1 1
192 30 =
1
5760
L
L
0
0
xyz dx dy dz = 1 L
xdx
L 3
0
L
0
ydy
L
0
zdz
= 1 L 3 x
2
2
L
0
y
2
2
L
0
z
2
2
L
0
= 1 L 3 L 2
2
L 2
2
L 2
2 = L3
8
53. The triple integral will attain its maximum when the integrand 1 − x 2 − 2y 2 − 3z 2 is positive in the region E and negative
everywhere else. For if E contains some region F where the integrand is negative, the integral could be increased by excluding
F from E,andifE fails to contain some part G of the region where the integrand is positive, the integral could be increased
by including G in E. Sowerequirethatx 2 +2y 2 +3z 2 ≤ 1. This describes the region bounded by the ellipsoid
x 2 +2y 2 +3z 2 =1.