Solução_Calculo_Stewart_6e

F.
SECTION 16.8TX.10
TRIPLE INTEGRALS IN SPHERICAL COORDINATES ET SECTION 15.8 ¤ 251
16.8 Triple Integrals in Spherical Coordinates ET 15.8
1. (a)
(b)
x = ρ sin φ cos θ =(1)sin0cos0=0,
y = ρ sin φ sin θ =(1)sin0sin0=0,and
z = ρ cos φ =(1)cos0=1so the point is
(0, 0, 1) in rectangular coordinates.
x =2sin π cos π = √ 2
, y 4 3 2 =2sinπ sin π = √ 6
,
4 3 2
z =2cos π = √ √2
2 so the point is , √ 6
, √
2 in
4 2 2
rectangular coordinates.
3. (a) ρ = x 2 + y 2 + z 2 = √ 1+3+12=4, cos φ = z ρ = 2 √ 3
4
cos θ =
x
ρ sin φ = 1
4sin(π/6) = 1 2
⇒ θ = π 3
=
√
3
2
⇒
φ = π 6 ,and
[since y>0]. Thus spherical coordinates are
4, π 3 , π
.
6
(b) ρ = √ 0+1+1= √ 2, cos φ = −1 √
2
⇒ φ = 3π 4 ,andcos θ = 0
√
2sin(3π/4)
=0 ⇒ θ = 3π 2
[since y