dx dx dt x

tance is assumed to be negligible, then its position after seconds is given by the parametric equations

where is the acceleration due to gravity ( ms ).

(a) If a gun is fired with and ms, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet?

; (b) Use a graphing device to check your answers to part (a).

Then graph the path of the projectile for several other values of the angle to see where it hits the ground.

Summarize your findings.

(c) Show that the path is parabolic by eliminating the parameter.

; Investigate the family of curves defined by the parametric equations , . How does the shape change as increases? Illustrate by graphing several members of the family.

;^{40. The}swallowtail catastrophe curvesare defined by the

parametric equations , .

Graph several of these curves. What features do the curves have in common? How do they change when increases?

; The curves with equations , are

calledLissajous figures. Investigate how these curves vary when , , and vary. (Take to be a positive integer.)

;^42. Investigate the family of curves defined by the parametric equations

How does the shape change as cchanges? In particular, you should identify the transitional values of cfor which the basic shape of the curve changes.

ycostcsint xsintcsint

n n

b a

ybcost xasinnt

41.

c yct²3t⁴ x2ct4t³

c

yt³ct xt²

39.

v0500 30

9.8 2

t

y^v0sint¹2tt² x^v0cost

t 36. A curve, called a witch of Maria Agnesi, consists of all

possible positions of the point in the figure. Show that parametric equations for this curve can be written as

Sketch the curve.

;^37. Suppose that the position of one particle at time is given by

and the position of a second particle is given by

(a) Graph the paths of both particles. How many points of intersection are there?

(b) Are any of these points of intersection collision points?

In other words, are the particles ever at the same place at the same time? If so, find the collision points.

(c) Describe what happens if the path of the second particle is given by

38. If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resis-

v0

x23cost y21sint 0t2 0t2 y21sint

x23cost

0t2 y12 cos t

x13 sin t

t O

A P

y=2a C

y

x a

¨

y2asin² x2acot

P

CALCULUS WITH PARAMETRIC CURVES

Having seen how to represent curves by parametric equations, we now apply the meth- ods of calculus to these parametric curves. In particular, we solve problems involving tangents, areas, and arc length.

TANGENTS

Suppose and are differentiable functions and we want to find the tangent line at a point on the parametric curve , where is also a differentiable func- tion of . Then the Chain Rule gives

dy dt dy

dx dx dt x

y y

t

t

x f t

t

f

9.2

If , we can solve for :

Equation 1 (which you can remember by thinking of canceling the ’s) enables us to find the slope of the tangent to a parametric curve without having to elimi- nate the parameter . We see from (1) that the curve has a horizontal tangent when

(provided that ) and it has a vertical tangent when (provided that ).

As we know from Chapter 4, it is also useful to consider . This can be found by replacing y by in Equation 1:

EXAMPLE 1

A curve is defined by the parametric equations , (a) Show that has two tangents at the point (3, 0) and find their equations.

(b) Find the points on where the tangent is horizontal or vertical.

(c) Determine where the curve is concave upward or downward.

(d) Sketch the curve.

SOLUTION

(a) Notice that when or . Therefore,

the point on arises from two values of the parameter, and . This indicates that crosses itself at . Since

the slope of the tangent when is , so the

equations of the tangents at are

(b) has a horizontal tangent when , that is, when and . Since , this happens when , that is, . The corresponding points on are and (1, 2). has a vertical tangent when

, that is, . (Note that there.) The corresponding point on is (0, 0).

(c) To determine concavity we calculate the second derivative:

d

²

y dx

²

d dt dy dx

dx dt

3

2 1

t 1

²

2t 3 t

²

1 4t

³

C

dy dt 0 t 0

dx dt 2t 0

1, 2 C C

t 1 t

²

1

dy dt 3t

²

3 dx dt 0

dy dt 0 dy dx 0

C

y

s

3 x 3 and

y

s

3 x 3

3, 0 t

s

3 dy dx 6 ( 2

s

3 )

_s

3 dy

dx dy dt

dx dt 3t

²

3

2t 3

2 t

1 t

3, 0 C

t

s

3 t

s

3

3, 0 y C t

³

3t t t

²

3 0 t 0 t

s

3

C C

y t

³

3t.

x t

²

C

d

²

y dx

²

d

dx dy dx dt d dx dy

dx dt dy dx

d

²

y dx

²

dy dt 0

dx dt 0 dx dt 0

dy dt 0

t dy dx

dt dy

dx

dy dt dx dt

if

dx dt 0

1

dy dx dx dt 0

■ If we think of the curve as being traced out by a moving particle, then

and are the vertical and horizontal velocities of the particle and Formula 1 says that the slope of the tangent is the ratio of these velocities.

dxdt dydt

| Note that ^d .

2y dx²

d²y dt² d²x dt²

Thus the curve is concave upward when and concave downward when . (d) Using the information from parts (b) and (c), we sketch in Figure 1.

■

EXAMPLE 2

(a) Find the tangent to the cycloid , at the point

where . (See Example 7 in Section 9.1.)

(b) At what points is the tangent horizontal? When is it vertical?

SOLUTION

(a) The slope of the tangent line is

When , we have

and

Therefore, the slope of the tangent is and its equation is

The tangent is sketched in Figure 2.

(b) The tangent is horizontal when , which occurs when and , that is, , an integer. The corresponding point on the

cycloid is .

When , both and are 0. It appears from the graph that there are vertical tangents at those points. We can verify this by using l’Hospital’s Rule as follows:

A similar computation shows that as , so indeed there are vertical tangents when 2n , that is, when dy dx

l

x 2nr

l

2n .

■

l

lim

2n

dy

dx lim

l2n

sin

1 cos lim

l2n

cos

sin

dy d dx d

( 2n 2n 1 r, 2r

)

2n

1

n

1 cos 0

sin 0 dy dx 0

0 y

x

2πr 4πr

(πr, 2r)

(_πr, 2r) (3πr, 2r) (5πr, 2r)

π

¨=3

FIGURE 2

s

3 x y r s3 2

or y r

2

s

3 x

r 3

r

^s

2 3

s

3 dy

dx sin 3

1 cos 3

s

3 2 1

¹2

s

3

y r 1

cos

3 2 r

x r 3 sin 3 r 3 s2 3

3 dy

dx dy d

dx d r sin

r 1 cos sin 1 cos 3

y r 1 cos

x r sin

V

C

t 0 t 0

0 y

x (3, 0)

(1, _2) (1, 2)

t=1 t=_1

y=œ„3(x-3)

y=_ œ„3(x-3) FIGURE 1

AREAS

We know that the area under a curve from to is , where . If the curve is given by parametric equations , and is tra- versed once as t increases from to , then we can adapt the earlier formula by using the Substitution Rule for Definite Integrals as follows:

EXAMPLE 3

Find the area under one arch of the cycloid , . (See Figure 3.)

SOLUTION

One arch of the cycloid is given by . Using the Substitution

Rule with and , we have

■

ARC LENGTH

We already know how to find the length of a curve given in the form , . Formula 7.4.3 says that if is continuous, then

Suppose that can also be described by the parametric equations , , , where . This means that is traversed once, from left to right, as increases from to and , . Putting Formula 1 into Formula 2 and using the Substitution Rule, we obtain

Since , we have

Even if can’t be expressed in the form , Formula 3 is still valid but we obtain it by polygonal approximations. We divide the parameter interval into n subintervals of equal width t . If , , , . . . , are the endpoints of these subinter- t

0

t

1

t

2

t

n

,

y F x C

L y

dx dt

²

dy dt

²

dt

3

dx dt 0

L y

a^b

1 dy dx 2dxy 1 dy dxdt dt 2 dx dt dt

f b

f a

t

C dx dt f t 0

t C x f t y

t

t

L y

a^b

1 dy dx 2dx

2

F a x b

y F x C

L

r

²

(

³2

2 ) 3 r

²

r

²

[

³2

2 sin

¹4

sin 2 ]

0

2

r

²

y

0²

[ 1 2 cos

¹2

1 cos 2 ] d

r

²

y

0²

1 cos

²

d r

²

y

0²

1 2 cos cos

²

d A y

0^2r

y dx y

0²

r 1 cos r 1 cos d

dx r 1 cos d

y r 1 cos 0 2

y

^V

r 1 cos x r sin

or yttftdt if ( f,t) is the leftmost endpoint

A y

a^b

y dx y

t

t f t dt

x f t y

t

t F x 0

A x

a^b

F x dx b

a y F x

FIGURE 3 0 y

x 2πr

■ The result of Example 3 says that the area under one arch of the cycloid is three times the area of the rolling circle that generates the cycloid (see Example 7 in Section 9.1). Galileo guessed this result but it was first proved by the French mathematician Roberval and the Italian mathematician Torricelli.

vals, then and are the coordinates of points that lie on and the polygon with vertices , , . . . , approximates (see Figure 4).

As in Section 7.4, we define the length of to be the limit of the lengths of these approximating polygons as :

The Mean Value Theorem, when applied to on the interval , gives a number

in such that

If we let and , this equation becomes

Similarly, when applied to , the Mean Value Theorem gives a number in such that

Therefore

and so

The sum in (4) resembles a Riemann sum for the function but it is not exactly a Riemann sum because in general. Nevertheless, if and are continuous, it can be shown that the limit in (4) is the same as if and were equal, namely,

Thus, using Leibniz notation, we have the following result, which has the same form as (3).

THEOREM

If a curve is described by the parametric equations , , , where and are continuous on and is tra- versed exactly once as increases from to , then the length of is

Notice that the formula in Theorem 5 is consistent with the general formulas and ds

²

dx

²

dy

²

of Section 7.4.

L x ds

L y

dx dt

²

dy dt

²

dt

C

t

, C

t

f

t

y

5t

t C x f t

L y

^s

f

t

^2t

t

²

dt

t

i

**

t

i

*

t

f t

i

* t

i

**

s

f t

²t

t

²

L lim

nl_i

ⁿ₁^s

f

t

ⁱ

*

^2t

t

ⁱ

**

²

t

4

s

f t

i

*

²t

t

_i

**

²

t

s

f t

i

* t

²t

t

_i

** t

²

P

ⁱ¹

P

ⁱ

^s

x

ⁱ²

y

ⁱ²

y

it

t

i

** t

t

i1

, t

i

t

i

**

t

x

i

f t

i

* t y

i

y

i

y

i1

x

i

x

i

x

i1

f t

i

f t

i1

f t

i

* t

i

t

i1

t

i1

, t

i

t

i

* f t

i1

, t

i

L lim

nl_i1

ⁿ

P

ⁱ¹

P

ⁱ

n

l

L C

C P

n

P

1

P

0

C P

i

x

i

, y

i

y

it

t

i

x

i

f t

i

P_i P_i-1

0 y

x P¸

P¡

P™

C

FIGURE 4

P_n

EXAMPLE 4

If we use the representation of the unit circle given in Example 2 in Section 9.1,

then and , so Theorem 5 gives

as expected. If, on the other hand, we use the representation given in Example 3 in Section 9.1,

then , , and the integral in Theorem 5 gives

|

Notice that the integral gives twice the arc length of the circle because as increases from 0 to , the point traverses the circle twice. In general, when finding the length of a curve from a parametric representation, we have to be careful to ensure that is traversed only once as increases from to .

■

EXAMPLE 5

Find the length of one arch of the cycloid ,

SOLUTION

From Example 3 we see that one arch is described by the parameter

interval . Since

we have

To evaluate this integral we use the identity with ,

which gives . Since , we have

and so . Therefore

and so

2r 2 2 8r

■

L 2r y

0²

sin 2 d 2r 2 cos 2 ]

0 2

s

2 1 cos

s

4 sin

²

2 2 sin

2

2 sin

2

sin 2 0

0 2

0 2

1 cos 2 sin

²

2 sin

²

x

¹2

1 cos 2x 2x y

0²s

r

²

1 2 cos cos

²

sin

²

d r y

0²s

2 1 cos d L y

0²

dx d

²

dy d

²

d

y

^02s

r

²

1

cos

²

r

²

sin

²

d

dy

d r sin dx and

d r 1 cos

0 2

y r 1 cos .

x r sin

V

t

C C sin 2t, cos 2t

2 t

y

0²

dx dt

²

dy dt

²

dt

y

^02s

4 cos

²

2t

4 sin

²

2t dt

y

⁰²

2 dt

4

dy dt 2 sin 2t dx dt 2 cos 2t

0 t 2

y cos 2t x sin 2t

y

0²

dt 2 L y

0²

dx dt

²

dy dt

²

dt

y

^02s

sin

²

t

cos

²

t dt

dy dt cos t dx dt sin t

0 t 2

y sin t x cos t

■ The result of Example 5 says that the length of one arch of a cycloid is eight times the radius of the generating circle (see Figure 5). This was first proved in 1658 by Sir Christopher Wren, who later became the architect of St. Paul’s Cathedral in London.

FIGURE 5 0 y

x 2πr L=8r r

20. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Show that the curve , has two tan-

gents at and find their equations. Sketch the curve.

22. At what point does the curve ,

cross itself ? Find the equations of both tangents at that point.

23. (a) Find the slope of the tangent line to the trochoid

, in terms of . (See

Exercise 34 in Section 9.1.)

(b) Show that if , then the trochoid does not have a vertical tangent.

24. (a) Find the slope of the tangent to the astroid , in terms of .

(b) At what points is the tangent horizontal or vertical?

(c) At what points does the tangent have slope 1 or ? 25. At what points on the curve , is the

tangent parallel to the line with equations ,

?

26. Find equations of the tangents to the curve , that pass through the point .

Use the parametric equations of an ellipse, , , , to find the area that it encloses.

28. Find the area bounded by the curve , and the line .

29. Find the area bounded by the curve , , , and the lines and . 30. Find the area of the region enclosed by the astroid

, .

31. Find the area under one arch of the trochoid of Exercise 34 in Section 9.1 for the case .

32. Let be the region enclosed by the loop of the curve in Example 1.

(a) Find the area of .

(b) If is rotated about the -axis, find the volume of the resulting solid.

(c) Find the centroid of .

33–36 ^■ Set up, but do not evaluate, an integral that represents the length of the curve.

33. , ,

34. , ,

35. xtcost, ytsint, 0t2

3t3

yt² x1e^t

1t2 y⁴3t³²

xtt²

᏾

᏾ ᏾ x

᏾

dr yasin³ xacos³

x0 y1

0t2

ye^t xcost y2.5

yt1t

xt1t

0 2

ybsin

xacos 27.

4, 3 y2t³1

x3t²1 y12t5

x7t y6t² xt³4t

1

yasin³

xacos³ dr

yrdcos

xr dsin

ytant12 cos²t

x12 cos²t 0, 0

ysint cost xcost

21.

y2t²t xt⁴4t³8t²

1–2 ^■ Find .

1. , 2. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

3–6 ^■ Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

3. , ;

4. , ;

, ;

6. , ;

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

7. Find an equation of the tangent to the curve , at the point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.

; ^8. Find equations of the tangents to the curve , at the origin. Then graph the curve and the tangents.

9–12 ^■ Find and . For which values of is the curve concave upward?

,

10. ,

11. ,

12. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

13–16 ^■ Find the points on the curve where the tangent is hori- zontal or vertical. If you have a graphing device, graph the curve to check your work.

13. ,

14. ,

15. ,

16. ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

;^17. Use a graph to estimate the coordinates of the leftmost point on the curve , . Then use calculus to find the exact coordinates.

;^18. Try to estimate the coordinates of the highest point and the leftmost point on the curve , . Then find the exact coordinates. What are the asymptotes of this curve?

;^{19–20 ■} Graph the curve in a viewing rectangle that displays all the important aspects of the curve.

, yt³t xt⁴2t³2t²

19.

yte^t xte^t

ytlnt xt⁴t²

y2 sin xcos 3

ysin 2 x2 cos

y2t³3t²1 x2t³3t²12t

yt³12t x10t²

ytlnt xtlnt

yte^t xte^t

yt²1 xt³12t

yt²t³ x4t²

9.

t d²ydx²

dydx ysintsint

xsint 1, 1

yt1²

xe^t 0 ysin cos 2

xcos sin 2

t1 ytlnt² xe^st

5.

t3 y¹3t³t x2t²1

t1 yt³t

xt⁴1

yte^t xte^t

y25t xtt³

dydx

EXERCISES

9.2

51. (a) Graph the epitrochoidwith equations

What parameter interval gives the complete curve?

(b) Use your CAS to find the approximate length of this curve.

52. A curve called Cornu’s spiralis defined by the parametric equations

where and are the Fresnel functions that were intro- duced in Chapter 5.

(a) Graph this curve. What happens as and as

?

(b) Find the length of Cornu’s spiral from the origin to the point with parameter value .

53. A string is wound around a circle and then unwound while being held taut. The curve traced by the point at the end of the string is called the involuteof the circle. If the circle has radius and center and the initial position of is

, and if the parameter is chosen as in the figure, show that parametric equations of the involute are

54. A cow is tied to a silo with radius by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow.

r O x

y

r

¨ P

T

yrsin cos xrcos sin

r, 0

P O

r

P t

tl

tl S

C

ySt

y

0^t sinu²2du xCt

y

0^t cosu²2du

CAS

y11 sin t4 sin11t2 x11 cos t4 cos11t2

36. , , CAS

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

37– 40 ^■ Find the length of the curve.

, ,

38. , ,

39. , ,

40. , ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

;^{41– 43 ■} Graph the curve and find its length.

, ,

42. , ,

43. , ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

44. Find the length of the loop of the curve , .

45. Use Simpson’s Rule with to estimate the length of

the curve , , .

46. In Exercise 36 in Section 9.1 you were asked to derive the

parametric equations , for the

curve called the witch of Maria Agnesi. Use Simpson’s Rule with to estimate the length of the arc of this curve

given by .

47– 48 ^■ Find the distance traveled by a particle with position as varies in the given time interval. Compare with the length of the curve.

47. , ,

48. , ,

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

49. Show that the total length of the ellipse ,

, , is

where is the eccentricity of the ellipse , where .

50. Find the total length of the astroid , , wherea0.

yasin³ xacos³

csae²b²

) (

eca

L4a

y

0²s1e²sin²d

ab0

ybcos xasin

0t4 ycost

xcos²t

0t3 ycos²t

xsin²t x,y t

4 2 n4

y2asin² x2acot

6t6

yte^t xte^t

n6 y3t²

x3tt³

8t3

y4e^t2 xe^tt

4t34 ysint

xcostln

(

tan¹2t

)

0t

ye^tsint xe^tcost

41.

0t3 y52t

xe^te^t

0t2 yln1t

x t

1t

0 yasin cos

xacos sin

0t1 y42t³

x13t² 37.

1t5 yst1

xlnt