9.4

;^65. A family of curves has polar equations

Investigate how the graph changes as the number changes.

In particular, you should identify the transitional values of for which the basic shape of the curve changes.

;^66. The astronomer Giovanni Cassini (1625 –1712) studied the family of curves with polar equations

where and are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of and . (Cassini thought that these curves might represent planetary orbits better than Kepler’s ellipses.) Investigate the variety of shapes that these curves may have. In particular, how are and related to each other when the curve splits into two parts?

67. Let be any point (except the origin) on the curve . If is the angle between the tangent line at and the radial line , show that

[Hint: Observe that in the figure.]

68. (a) Use Exercise 67 to show that the angle between the tan- gent line and the radial line is at every point on the curve .

; (b) Illustrate part (a) by graphing the curve and the tangent lines at the points where and .

(c) Prove that any polar curve with the property that the angle between the radial line and the tangent line is a constant must be of the form , where and are k constants.

C r Ce^k

#

r f 2

  0 r e^ #  4 O

P ÿ

¨ ˙

r=f(¨ )

#     tan#  r

drd

OP

# P

r f 

P

c a c a c

a

r⁴ 2c²r²cos 2  c⁴ a⁴ 0

a a r 1 a cos

1 a cos

, 50. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

51–54 ^■ Find the points on the given curve where the tangent line is horizontal or vertical.

52.

53. 54.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Show that the polar equation , where

, represents a circle, and find its center and radius.

56. Show that the curves and intersect at right angles.

;^{57–60 ■} Use a graphing device to graph the polar curve.

Choose the parameter interval to make sure that you produce the entire curve.

57. (butterfly curve)

58.

59. 60.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

;^61. How are the graphs of and

related to the graph of ? In general, how is the graph of related to the

graph of ?

;^62. Use a graph to estimate the -coordinate of the highest points on the curve . Then use calculus to find the exact value.

;^63. (a) Investigate the family of curves defined by the polar equations , where is a positive integer. How is the number of loops related to ?

(b) What happens if the equation in part (a) is replaced by

?

;^64. A family of curves is given by the equations

, where is a real number and is a posi- tive integer. How does the graph change as increases?

How does it change as changes? Illustrate by graphing enough members of the family to support your conclusions.

c

n n c

r 1  c sin n

r





n n r sin n

r sin 2y r f 

r f   r 1  sin r 1  sin  3

r 1  sin  6

r cos2  cos3

r 2  5 sin6

r sin²4  cos4

r e^sin 2 cos4

r a cos

r a sin ab 0

r a sin  b cos  55.

r² sin 2

r 1  cos

r e^ r 3 cos 

51.

  6 r sin 3

   r 1

49.

AREAS AND LENGTHS IN POLAR COORDINATES

In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the area of a sector of a circle

where, as in Figure 1, is the radius and is the radian measure of the central angle. r  A¹2r²

1

9.4

¨ r

FIGURE 1

Formula 1 follows from the fact that the area of a sector is proportional to its central angle: . (See also Exercise 67 in Section 6.2.)

Let be the region, illustrated in Figure 2, bounded by the polar curve

and by the rays and , where is a positive continuous function and where . We divide the interval into subintervals with endpoints , , , . . . , and equal width . The rays then divide into smaller regions with central angle . If we choose in the th subinterval , then the area of the th region is approximated by the area of the sector of a circle with central angle and radius . (See Figure 3.)

Thus from Formula 1 we have

and so an approximation to the total area of is

It appears from Figure 3 that the approximation in (2) improves as . But the sums in (2) are Riemann sums for the function , so

It therefore appears plausible (and can in fact be proved) that the formula for the area of the polar region is

Formula 3 is often written as

with the understanding that . Note the similarity between Formulas 1 and 4.

When we apply Formula 3 or 4 it is helpful to think of the area as being swept out by a rotating ray through that starts with angle and ends with angle .

EXAMPLE 1 Find the area enclosed by one loop of the four-leaved rose .

SOLUTION The curve was sketched in Example 8 in Section 9.3. Notice from Figure 4 that the region enclosed by the right loop is swept out by a ray that rotates from to . Therefore, Formula 4 gives

¹2

[

]

4 



y

1

21  cos 4 d

A

y

y

y

  4

  4

r cos 2

r cos 2

V

b a

O

r f 

A

y

4

A

y

2 f  ²d

3

᏾ A

nl lim_i



y

2 f  ²d

t ¹2 f  ² nl  A _i



2

᏾ A

A^{i 1}2 f ⁱ* ²

fⁱ*

 i

Aⁱ

ⁱ1,ⁱ   ⁱⁱ1 ⁱ* i

᏾ n

  ⁱ



ⁿ

²

¹ a, b ⁰

0 b  a 2

  b f

  a r f 

᏾

A 2r²¹2r²

FIGURE 2 O

¨=b

b ¨=a

r=f(¨)

a

᏾

O

¨=b

¨=a

¨=¨_i-1

¨=¨_i

Ψ

f(¨_i^*)

FIGURE 3

r=cos 2¨ ¨=^π₄

¨=_^π₄

FIGURE 4

EXAMPLE 2 Find the area of the region that lies inside the circle and outside the cardioid .

SOLUTION The cardioid (see Example 7 in Section 9.3) and the circle are sketched in Figure 5 and the desired region is shaded. The values of and in Formula 4 are determined by finding the points of intersection of the two curves. They intersect

when , which gives , so , . The desired

area can be found by subtracting the area inside the cardioid between and from the area inside the circle from to . Thus

Since the region is symmetric about the vertical axis , we can write

[because ]

■

Example 2 illustrates the procedure for finding the area of the region bounded by two polar curves. In general, let be a region, as illustrated in Figure 6, that is bounded by curves with polar equations , , , and , where

and . The area of is found by subtracting the area inside from the area inside , so using Formula 3 we have

| ^CAUTION The fact that a single point has many representations in polar coordinates sometimes makes it difficult to find all the points of intersection of two polar curves.

For instance, it is obvious from Figure 5 that the circle and the cardioid have three points of intersection; however, in Example 2 we solved the equations and and found only two such points, and . The origin is also a point of intersection, but we can’t find it by solving the equations of the curves because the origin has no single representation in polar coordinates that satisfies both equations. Notice that, when represented as or , the origin satisfies and so it lies on the circle; when represented as , it satisfies and so it lies on the cardioid. Think of two points moving along the curves as the parameter value increases from 0 to . On one curve the origin is reached at and ; on the other curve it is reached at . The points don’t collide at the origin because they reach the origin at different times, but the curves intersect there nonetheless.

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 cal- culator or computer to help with this task.

  32

  

  0  2

r 1  sin 0, 32

r 3 sin  0, 0 0,

(

) (

)

r 1  sin r 3 sin

¹2

y

(

)

y

1

2 f  ²d 

y

2t ²d

r f 

r t 0 b  a 2 A ᏾

f  t  0 ᏾ r f  r t   a   b

 3  2 sin 2  2 cos 

]

2

sin² ¹21  cos 2



y



y

A 2



^y

^y



A¹2

y

y

56

6

  563 sin   1  sin  sin ¹2   6 56   6 b

a

r 1  sin r 3 sin 

V

FIGURE 5 O

¨=^5π₆ ¨=^π₆

r=3 sin ¨

r=1+sin ¨

O

¨=b

¨=a

r=f(¨)

᏾ r=g(¨)

FIGURE 6

EXAMPLE 3 Find all points of intersection of the curves and . SOLUTION If we solve the equations and , we get and, therefore,  , , , . Thus the values of between 0 and that satisfy both equations are , , , . We have found four

points of intersection: , , and .

However, you can see from Figure 7 that the curves have four other points of

intersection—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

so, using , we have

Assuming that is continuous, we can use Formula 9.2.5 to write the arc length as

Therefore, the length of a curve with polar equation , , is

EXAMPLE 4 Find the length of the cardioid .

SOLUTION The cardioid is shown in Figure 8. (We sketched it in Example 7 in Section 9.3.) Its full length is given by the parameter interval 0  2, so

r 1  sin

V

L

y







5

a  b r f 

L

y









f

























dy d  dr

d sin   r cos  dx

d  dr

d cos   r sin 



y r sin  f  sin  x r cos  f  cos 

 a  b

r f 

r ¹2

r cos 2 r ¹2

(

) (

)

(

) (

)

(

) (

)

(

)

(

)

2

 113

73 53

3

2 r cos 2 r¹2 cos 2 ¹2

r¹2

r cos 2

FIGURE 7 r=cos 2¨

1 r=2

”   ,     ’¹₂ ^π₃

”   ,    ’¹₂^π₆

O

FIGURE 8 r=1+sin ¨

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 L 8. _■

s2  2 sin 



y

L

y







^y

15–18 ^■ Find the area of the region enclosed by one loop of the curve.

15. 16.

(inner loop) 18.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

19–22 ^■ Find the area of the region that lies inside the first curve and outside the second curve.

19. ,

20. ,

,

22. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

23–26 ^■ Find the area of the region that lies inside both curves.

23. , 24. ,

, 26. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

27. Find the area inside the larger loop and outside the smaller loop of the limaçon .

28. When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the micro- phone is placed 4 m from the front of the stage (as in the figure) and the boundary of the optimal pickup region is

stage

audience

microphone 12 m

4 m r¹2 cos

r 1 r² 2 sin 2

r cos 2

r sin 2

25.

r sin r sin 2

r cos r sin

r 3 sin  r 2  sin

r 1  cos r 3 cos 

21.

r 1 r 1  sin

r 2 r 4 sin

r 2 cos   sec  r 1  2 sin

17.

r 4 sin 3

r sin 2

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.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

9–12 ^■ Sketch the curve and find the area that it encloses.

10.

11. 12.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

;^{13–14 ■} Graph the curve and find the area that it encloses.

13. 14.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

r 2 sin   3 sin 9

r 1  2 sin 6

r 2  cos 2

r 2 cos 3

r 31  cos

r² 4 cos 2

9.

r=sin 4¨

r=4+3 sin ¨ 7.

r=¨ r=1+sin ¨

0   r ssin 

3  23 r sin

  2

r e^2

0  4 r s

EXERCISES

9.4

33–36 ^■ Find the exact length of the polar curve.

33. ,

34. ,

,

36. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

37–38 ^■ Use a calculator to find the length of the curve correct to four decimal places.

37. 38.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

r 4 sin 3

r 3 sin 2

0  2

r

0  2

r² 35.

0  2

r e^2

0  3 r 3 sin 

given by the cardioid , 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.

29–32 ^■ Find all points of intersection of the given curves.

29. ,

30. ,

,

32. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

r² cos 2

r² sin 2

r sin 2

r sin 31.

r sin 3

r cos 3

r 1  cos r cos

r r 8  8 sin 

CONIC SECTIONS IN POLAR COORDINATES

In your previous study of conic sections, parabolas were defined in terms of a focus and directrix whereas ellipses and hyperbolas were defined in terms of two foci. After reviewing those definitions and equations, we present a more unified treatment of all three types of conic sections in terms of a focus and directrix. Furthermore, if we place the focus at the origin, then a conic section has a simple polar equation. In Chapter 10 we will use the polar equation of an ellipse to derive Kepler’s laws of planetary motion.

CONICS IN CARTESIAN COORDINATES

Here we provide a brief reminder of what you need to know about conic sections. A more thorough review can be found on the website www.stewartcalculus.com.

Recall that a parabola is the set of points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This definition is illustrated by Figure 1. Notice that the point halfway between the focus and the directrix lies on the parabola; it is called the vertex. The line through the focus per- pendicular to the directrix is called the axis of the parabola.

A parabola has a very simple equation if its vertex is placed at the origin and its directrix is parallel to the -axis or -axis. If the focus is on the -axis at the point , then the directrix has the equation and an equation of the parabola is . [See parts (a) and (b) of Figure 2.] If the focus is on the -axis at , then the directrix is and an equation is as in parts (c) and (d).

(d) ¥=4px, p<0 (c) ¥=4px, p>0

(b) ≈=4py, p<0 (a) ≈=4py, p>0

FIGURE 2

0 x

y

( p, 0)

x=_p

0 x

y

( p, 0)

x=_p 0

x y

(0, p) y=_p

0 x

y

(0, p)

y=_p

y² 4px

x p x  p, 0

x² 4py

y p

0, p

y y

x F

9.5

axis

focus F

parabola

directrix FIGURE 1

vertex