Problem 3.25 Use the appropriate expression for the differential surface area dsto
determine the area of each of the following surfaces:
(a) r=3; 0 ≤
φ
≤
π
/3; −2≤z≤2,
(b) 2≤r≤5;
π
/2≤
φ
≤
π
;z=0,
(c) 2≤r≤5;
φ
=
π
/4; −2≤z≤2,
(d) R=2; 0 ≤
θ
≤
π
/3; 0 ≤
φ
≤
π
,
(e) 0≤R≤5;
θ
=
π
/3; 0 ≤
φ
≤2
π
.
Also sketch the outlines of each of the surfaces.
Solution:
32
∆Φ = π/3
25
y
x
25
(a) (b)
(d) (e)
(c)
Figure P3.25: Surfaces described by Problem 3.25.
(a) Using Eq. (3.43a),
A=Z2
z=−2Z
π
/3
φ
=0(r)|r=3d
φ
dz =³(3
φ
z)|
π
/3
φ
=0´¯¯¯
2
z=−2=4
π
.
(b) Using Eq. (3.43c),
A=Z5
r=2Z
π
φ
=
π
/2(r)|z=0d
φ
dr =³¡1
2r2
φ
¢¯¯
5
r=2´¯¯¯
π
φ
=
π
/2=21
π
4.
