Quiz11 solution
1. Evaluate the line integral
(0, 0, 0) to (1, 2, 3).
R
C
xeyz ds if C is the line segment from
sol:
r(t) = (t, 2t, 3t), r′ (t) = (1, 2, 3), 0 ≤ t ≤ 1
p
√
ds
= |r′ (t)| = (1)2 + (2)2 + (3)2 = 14
dt
Z
yz
xe
Z
1
2
te6t ·
ds =
√
14 dt
0
C
Z
6
=
0
√
1 u
e du
12
6
14 u
=
e
12
0
√
14 6
=
[e − 1].
12
1
R
2. Consider the surface intefral S z 2 dS, where S is the part of the
paraboloid x = y 2 + z 2 given by 0 ≤ x ≤ 1.
(1)write down a parametriz equation of S.
(2)Use the parametrization in (1) to evaluate the given surface integral.
sol:
(1).parametric equation S:
x = r2 , y = r cos θ, z = r sin θ, 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π.
(2).
u(r, θ) = (r2 , r cos θ, r sin θ)
ur = (2r, cos θ, sin θ), uθ = (0, −r sin θ, r cos θ)
Z
ZZ
2π
Z
f (u(r, θ)) |ur × uθ | dA =
D
0
0
1
r2 sin2 θ ·
√
r2 + 4r4 drdθ
√
1
(1 + 25 5)].
=π·[
120
2