1) QUIZ #6 -­ SOLUTIONS Use the Divergence Theorem to calculate the surface integral 
 
F
F
⋅
d
S
, w
here ( x, y, z ) = x 4iˆ − x 3z 2 ĵ + 4xy2 z k̂ , and S is the ∫∫
S
surface of the solid bounded by the cylinder x 2 + y 2 = 1 and the planes z = x + 2 and z = 0. 
∇ ⋅ F = 4x 3 + 4xy 2
2 π 1 r cosθ +2
 
2
2
3
∫∫ F ⋅ dS = ∫∫∫ 4x x + y dV = ∫ ∫ ∫ 4r cosθ r dz dr dθ = (
S
)
E
2π 1
∫ ∫ ( 4r
5
(
0 0
)
0
)
cos 2 θ + 8r 4 cosθ dr dθ =
0 0
2π
∫
0
2) Use the Divergence Theorem to verify the identity 
1  
Vol ( E ) = ∫∫ F ⋅ dS , where F ( x, y, z ) = xiˆ + yĵ + zk̂ . 3 S

1   1
1
F
⋅
d
S
=
div
F
dV = ∫∫∫ 3dV =
∫∫
∫∫∫
3 S
3 E
3 E
∫∫∫ dV = Vol ( E )
E
2
8
2
cos 2 θ + cosθ dθ = π 3
5
3