(inner loop)
22.
Find the area enclosed by the loop of the
strophoid
.
23–28
Find the area of the region that lies inside the first curve
and outside the second curve.
23.
,
24.
,
25.
,
26.
,
,
28.
,
29–34
Find the area of the region that lies inside both curves.
29.
,
30.
,
,
32.
,
33.
,
34.
,
,
,
35.
Find the area inside the larger loop and outside the smaller loop
of the limaçon
.
36.
Find the area between a large loop and the enclosed small loop
of the curve
.
37–42
Find all points of intersection of the given curves.
37.
,
38.
,
39.
,
40.
,
,
42.
,
r
2
cos 2
r
2
sin 2
r
sin 2
r
sin
41.
r
sin 3
r
cos 3
r
1
r
2 sin 2
r
1
sin
r
1
cos
r
3 sin
r
1
sin
r
1
2 cos 3
r
1
2
cos
b
0
a
0
r
b
cos
r
a
sin
r
2
cos 2
r
2
sin 2
r
3
2 sin
r
3
2 cos
r
cos 2
r
sin 2
31.
r
1
cos
r
1
cos
r
sin
r
s
3
cos
r
2
sin
r
3 sin
r
1
cos
r
3 cos
27.
r
3 sin
r
2
sin
r
2
r
2
8 cos 2
r
1
r
1
sin
r
1
r
2 cos
r
2 cos
sec
r
1
2 sin
21.
r
2 sin 6
r
3 cos 5
1–4
Find the area of the region that is bounded by the given curve
and lies in the specified sector.
1.
,
2.
,
3.
,
4.
,
5–8
Find the area of the shaded region.
5.
6.
8.
9–14
Sketch the curve and find the area that it encloses.
9.
10.
12.
13.
14.
;
15–16
Graph the curve and find the area that it encloses.
15.
16.
17–21
Find the area of the region enclosed by one loop of
the curve.
17.
18.
r
4 sin 3
r
sin 2
r
2 sin
3 sin 9
r
1
2 sin 6
r
2
cos 2
r
2 cos 3
r
2
sin
r
2
4 cos 2
11.
r
3 1
cos
r
3 cos
r=
sin
2¨
r=4+3
sin
¨
7.
r=1+
cos
¨
r=oe
„
¨
0
r
s
sin
3
2
3
r
sin
2
r
e
2
0
4
r
2
EXERCISES
10.4

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49–52
Use a calculator to find the length of the curve correct to
four decimal places.
49.
50.
51.
52.
;
53–54
Graph the curve and find its length.
53.
54.
55.
(a)
Use Formula 10.2.7 to show that the area of the surface
generated by rotating the polar curve
(where
is continuous and
) about the
polar axis is
(b) Use the formula in part (a) to find the surface area gener-
ated by rotating the lemniscate
about the
polar axis.
56.
(a)
Find a formula for the area of the surface generated by
rotating the polar curve
,
(where
is
continuous and
), about the line
.
(b) Find the surface area generated by rotating the lemniscate
about the line
.
2
r
2
cos 2
2
0
a
b
f
a
b
r
f
r
2
cos 2
S
y
b
a
2
r
sin
r
2
dr
d
2
d
0
a
b
f
a
b
r
f
r
cos
2
2
r
cos
4
4
r
1
cos
3
r
sin
2
r
4 sin 3
r
3 sin 2
;
43.
The points of intersection of the cardioid
and
the spiral loop
,
, can’t be found
exactly. Use a graphing device to find the approximate values
of
at which they intersect. Then use these values to estimate
the area that lies inside both curves.
44.
When recording live performances, sound engineers often use
a microphone with a cardioid pickup pattern because it sup-
presses noise from the audience. Suppose the microphone is
placed 4 m from the front of the stage (as in the figure) and
the boundary of the optimal pickup region is given by the car-
dioid
, where
is measured in meters and the
microphone is at the pole. The musicians want to know the
area they will have on stage within the optimal pickup range
of the microphone. Answer their question.