Math 2500
Fall 2009
Exam 3  Practice Questions
1. Evaluate the given line integral.
(a)
Z
C
yz
2
ds
,
where
C
is the line segment from (

1
,
1
,
3) to (0
,
3
,
5)
(b)
Z
C
x
3
z ds
,
where
C
:
x
= 2 sin
t, y
=
t, z
= 2 cos
t,
0
≤
t
≤
π/
2
(c)
Z
C
x
3
y dx

x dy
,
where
C
is the circle
x
2
+
y
2
= 1 with counterclockwise orientation
(d)
Z
C
x
sin
y dx
+
xyz dz
,
where
C
is given by
r
(
t
) =
t
i
+
t
2
j
+
t
3
k
, 0
≤
t
≤
1
(e)
Z
C
F
·
d
r
,
where
F
(
x, y
) =
x
2
y
i
+
e
y
j
and
C
is given by
r
(
t
) =
t
2
i

t
3
j
, 0
≤
t
≤
1
(f)
Z
C
F
·
d
r
,
where
F
(
x, y, z
) =
h
x
+
y, z, x
2
y
i
and
C
is given by
r
(
t
) =
h
2
t, t
2
, t
4
i
, 0
≤
t
≤
1
2. Find the work done by the force field
F
(
x, y, z
) =
z
i
+
x
j
+
y
k
in moving a particle from the
point (3
,
0
,
0) to the point (0
, π/
2
,
3)
(a) along a straight line
(b) along the helix
x
= 3 cos
t
,
y
=
t
,
z
= 3 sin
t
3. Show that
F
is a conservative vector field and find a potential function
f
such that
F
=
∇
f
.
(a)
F
(
x, y
) = sin
y
i
+ (
x
cos
y
+ sin
y
)
j
(b)
F
(
x, y, z
) = (2
xy
3
+
z
2
)
i
+ (3
x
2
y
2
+ 2
yz
)
j
+ (
y
2
+ 2
xz
)
k
4. Show that
F
is a conservative vector field and use this fact to evaluate
Z
C
F
·