Thus, to find
all
points of intersection of two polar curves, it is recommended that you
draw the graphs of both curves. It is especially convenient to use a graphing calculator or
computer to help with this task.
Find all points of intersection of the curves
and
.
SOLUTION
If we solve the equations
and
, we get
and, there-
fore,
,
,
,
. Thus the values of
between
and
that satisfy
both equations are
,
,
,
. We have found four points of inter-
section:
,
, and
.
However, you can see from Figure 7 that the curves have four other points of inter-
section—namely,
,
,
, and
. These can be found using
symmetry or by noticing that another equation of the circle is
and then solving
the equations
and
.
Arc Length
To find the length of a polar curve
,
, we regard
as a parameter and
write the parametric equations of the curve as
Using the Product Rule and differentiating with respect to
, we obtain
r
f
r
t
a
b
f
t
0
0
b
a
2
A
r
t
r
f
A
y
b
a
1
2
f
2
d
y
b
a
1
2
t
2
d
1
2
y
b
a
(
f
2
t
2
)
d
r
3 sin
r
1
sin
(
3
2
,
6
)
(
3
2
, 5
6
)
0, 0
0,
r
3 sin
0, 3
2
r
1
sin
2
0
3
2
r
cos 2
r
1
2
r
cos 2
r
1
2
cos 2
1
2
2
3 5
3 7
3 11
3
0
2
6 5
6 7
6 11
6
(
1
2
,
6
) (
1
2
, 5
6
)
,
(
1
2
, 7
6
)
(
1
2
, 11
6
)
(
1
2
,
3
) (
1
2
, 2
3
) (
1
2
, 4
3
)
(
1
2
, 5
3
)
r
1
2
r
cos 2
r
1
2
EXAMPLE 3
r
f
a
b
x
r
cos
f
cos
y
r
sin
f
sin
dx
d
dr
d
cos
r
sin
dy
d
dr
d
sin
r
cos
O
¨=b
¨=a
r=f(¨)
r=g(¨)
FIGURE 6
FIGURE 7
r=
cos
2¨
1
2
r=
” , ’
1
2
π
3
” , ’
1
2
π
6

##### We have textbook solutions for you!

**The document you are viewing contains questions related to this textbook.**

**The document you are viewing contains questions related to this textbook.**

Expert Verified

692
CHAPTER 10
so, using
, we have
Assuming that
is continuous, we can use Theorem 10.2.5 to write the arc length as
Therefore the length of a curve with polar equation
,
, is
Find the length of the cardioid
.
SOLUTION
The cardioid is shown in Figure 8. (We sketched it in Example 7 in
Section 10.3.) Its full length is given by the parameter interval
, so
Formula 5 gives
We could evaluate this integral by multiplying and dividing the integrand by
, or we could use a computer algebra system. In any event, we find that the
length of the cardioid is
.
dr
d
2
sin
2
2
r
dr
d
sin
cos
r
2
cos
2
dr
d
2
r
2
f
L
y
b
a
dx
d
2
dy
d
2
d
cos
2
sin
2
1
dx
d
2
dy
d
2
dr
d
2
cos
2
2
r
dr
d
cos
sin
r
2
sin
2
r
f
a
b
5
L
y
b
a
r
2
dr
d
2
d
r
1
sin
0
2
L
y
2
0
r
2
dr
d
2
d
y
2
0
s
1
sin
2
cos
2
d
y
2
0
s
2
2 sin
d
s
2
2 sin
L
8
v
EXAMPLE 4
O
FIGURE 8
r=1+
sin
¨
;
Graphing calculator or computer required
1.
Homework Hints available at stewartcalculus.com
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.
,
r
e
4
2
r
cos
0
6
r
2
9 sin 2
0
2
r
tan
6
3
r
0
5–8
Find the area of the shaded region.
5.
6.
r=œ
„
¨
r=1+
cos
¨
10.4
Exercises

AREAS AND LENGTHS IN POLAR COORDINATES
693
7.
8.
9–12
Sketch the curve and find the area that it encloses.
9.
10.
11.
12.
;
13–16
Graph the curve and find the area that it encloses.
13.
14.
15.
16.
17–21
Find the area of the region enclosed by one loop of
the curve.