S OLUTIONS TO M IDTERM 2
1). A helical wire follows the path c(t) = (3 cos(t), 3 sin(t), 4t) for 0 ≤ t ≤ 5π. Its
mass density λ (mass per unit length) is given by λ(x, y, z) = 2z. Find the mass
of the wire.
Solution: To find the density, we need to compute the path integral of λ(x, y, z)
over c(t). By definition, this is
Z
Z
5π
p
2(4t) (−3 sin(t))2 + (3 cos(t))2 + (4)2 dt
λds =
0
C
Z
= 40
0
5π
2 5π
t t dt = 40
= 500π 2 .
2 0
2). Let S be the parabolic surface given by z = 9 − x2 − y 2 for x2 + y 2 ≤ 9.
(a) Find a parametrization Φ : D → S. Be sure to specify the domain D.
Solution: There are a couple of parametrizations we could use here. The
most obvious is the graph parametrization
Φ(x, y) = (x, y, 9 − x2 − y 2 ), where (x, y) lie in the disk of radius 3.
We could also compose the parametrization with polar coordinates to obtain
Φ(r, θ) = (r cos(θ), r sin(θ), 9 − r2 ) where 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π.
(b) Use your parametrization to find a normal vector to S at the point (1, 1, 7).
Solution: Using the graph parametrization above with g(x, y) = 9 − x2 − y 2 ,
we have that
Tx × Ty = −
∂g
∂g
i−
j + k = 2xi + 2yj + k
∂x
∂y
gives a normal for a given (x, y). So the normal at (1, 1, 7) is
(Tx × Ty )(1, 1) = 2i + 2j + k.
3). The conical surface S given by x2 +y 2 = (3−z)2 with z ≥ 0 can be parametrized
by
Φ : [0, 2] × [0, 2π] → S ⊂ R3
Φ(r, θ) = (r cos(θ), r sin(θ), 3 − r).
Evaluate
RR
S
F · dS, where F is the radial vector field F(x, y, z) = (x, y, z).
1
2
Solution: By definition,
Z 2π Z 2
F (Φ(r, θ)) · (Tr × Tθ ) drdθ
F · dS = ±
0
0
S

 

Z 2π Z 2 r cos(θ)
Z 2π Z 2
r cos(θ)




r sin(θ) · r sin(θ)
=±
drdθ = ±
(r2 + 3r − r2 ) drdθ
0
0
0
0
3−r
r
Z 2
= ±2π
3r dr = ±12π.
Z Z
0
where the plus/minus above is included because no orientation of S is given. If
S is given the upward orientation, then
Z Z
F · dS = 12π
S
given that Tr × Tθ points upward (r > 0 is the k component of Tr × Tθ , which
implies that this vector points upward). If S if given the downward orientation,
then
Z Z
F · dS = −12π.
S
4). Green’s Theorem will not be covered on the exam. If you are curious how to do
this problem, refer to your lecture notes from 2/27/13.