14-51
SECTION 14.5
.
.
Curl and Divergence
1165
16.
C
(
y
sec
2
x
−
2)
dx
+
(tan
x
−
4
y
2
)
dy
, where
C
is formed by
x
=
1
−
y
2
and
x
=
0
17.
C
x
2
dx
+
2
x dy
+
(
z
−
2)
dz
, where
C
is the triangle from
(0
,
0
,
2) to (2
,
0
,
2) to (2
,
2
,
2) to (0
,
0
,
2)
18.
C
4
y dx
+
y
3
dy
+
z
4
dz
, where
C
is
x
2
+
y
2
=
4 in the plane
z
=
0
19.
C
F
·
d
r
, where
F
=
x
3
−
y
4
,
e
x
2
+
z
2
,
x
2
−
16
y
2
z
2
and
C
is
x
2
+
z
2
=
1 in the plane
y
=
0
20.
C
F
·
d
r
, where
F
=
x
3
−
y
2
z
,
√
x
2
+
z
2
,
4
xy
−
z
4
and
C
is
formed by
z
=
1
−
x
2
and
z
=
0 in the plane
y
=
2
In exercises 21–26, use a line integral to compute the area of the
given region.
21.
The ellipse 4
x
2
+
y
2
=
16
22.
The ellipse 4
x
2
+
y
2
=
4
23.
The region bounded by
x
2
/
3
+
y
2
/
3
=
1. (Hint: Let
x
=
cos
3
t
and
y
=
sin
3
t
)
24.
The region bounded by
x
2
/
5
+
y
2
/
5
=
1
25.
The region bounded by
y
=
x
2
and
y
=
4
26.
The region bounded by
y
=
x
2
and
y
=
2
x
27.
Use Green’s Theorem to show that the center of mass of the
region bounded by the positive curve
C
with constant den-
sity is given by ¯
x
=
1
2
A
C
x
2
dy
and ¯
y
= −
1
2
A
C
y
2
dx
, where
A
is the area of the region.
28.
Use the result of exercise 27 to find the center of mass of the
region in exercise 26, assuming constant density.
29.
Use the result of exercise 27 to find the center of mass of the
region bounded by the curve traced out by
t
3
−
t
,
1
−
t
2
, for
−
1
≤
t
≤
1, assuming constant density.
30.
Use the result of exercise 27 to find the center of mass of the
region bounded by the curve traced out by
t
2
−
t
,
t
3
−
t
, for
0
≤
t
≤
1, assuming constant density.
31.
Use Green’s Theorem to prove the change of variables formula
R
dA
=
S
∂
(
x
,
y
)
∂
(
u
,v
)
du d
v,
where
x
=
x
(
u
,v
) and
y
=
y
(
u
,v
) are functions with continu-
ous partial derivatives.
32.
For
F
=
1
x
2
+
y
2
−
y
,
x
and
C
any circle of radius
r
>
0 not
containing the origin, show that
C
F
·
d
r
=
0.
In exercises 33–36, use the technique of example 4.5 to evaluate
the line integral.
33.
C
F
·
d
r
, where
F
=
x
x
2
+
y
2
,
y
x
2
+
y
2
and
C
is any posi-
tively oriented simple closed curve containing the origin
34.
C
F
·
d
r
, where
F
=
y
2
−
x
2
(
x
2
+
y
2
)
2
,
−
2
xy
(
x
2
+
y
2
)
2
and
C
is any
positively oriented simple closed curve containing the origin
35.
C
F
·
d
r
, where
F
=
x
3
x
4
+
y
4
,
y
3
x
4
+
y
4
and
C
is any posi-
tively oriented simple closed curve containing the origin
36.
C
F
·
d
r
, where
F
=
y
2
x
x
4
+
y
4
,
−
x
2
y
x
4
+
y
4
and
C
is any posi-
tively oriented simple closed curve containing the origin
37.
Where is
F
(
x
,
y
)
=
2
x
x
2
+
y
2
,
2
y
x
2
+
y
2
defined? Show that
M
y
=
N
x
everywhere the partial derivatives are defined. If
C
is a simple closed curve enclosing the origin, does Green’s
Theorem guarantee that
C
F
·
d
r
=
0? Explain.
38.
For the vector field of exercise 37, show that
C
F
·
d
r
is the
same for all closed curves enclosing the origin.
39.
If
F
(
x
,
y
)
=
2
x
x
2
+
y
2
,
2
y
x
2
+
y
2
and
C
is a simple closed
curve in the fourth quadrant, does Green’s Theorem guarantee
that
C
F
·
d
r
=
0? Explain.