第6.3节 曲线长度

6.3 LENGTH OF A CRUVE

A segment of a curve in the plane (Figure 6.3.1) is described by

y= f(x), a≤ x≤ b.

What is its length? As usual, we shall give a definition and then justify it. A curve y= f(x) is said to be smooth if its derivative f '(x) is continuous. Our definition will assign a length to a segment of a smooth curve.

DEFINITION

Assume the function y=f(x) has a continuous derivative for x in [a, b], that is , the curve

y= f(x), a≤x≤b

is smooth. The length of the curve is defined as

Because ____________= ______________, the equation is sometimes

written in the form _________________

with the understanding that x is the independent variable. The length s is always greater than or equal to 0 because a < b and

____________ > 0.

JUSTIFICATION Let s (u,w) be the intuitive length of the curve between t = u and t = w.

The function s(u,w) has the Addition Property; the length of the curve from u to w equals

the length from u to v plus the length from v to w. Figure 6.3.2 shows an infinitesimal

piece of the curve from x to x + Δx. Its length is Δs = s (x, x+ Δx).

The slope dy/ dx is a continuous function of x, and therefore changes only by an infinitesimal amount between x and x+Δy. Hence

Δs ≈________ ( compared to Δx).

Dividing by Δx,

Then _________________ (compared to Δx).

Using the Infinite Sum Theorem,

EXAMPLE 1 Find the length of the curve

y=2x 3/2, 0 ≤ x≤ 1

Shown in Figure 6.3.3. We have

dy/dx = 3x1/2, _________________

Put u=1+9x. Then

___________________

Figure 6.3.3

Sometimes a curve in the (x,y) plane is given by parametric equations

x=f(t), y=g(t), c ≤ t ≤ d.

A natural example is the path of a moving particle where is time. We give a formula for the length of such a curve.

DEFINITION

Suppose the functions

x = f(t), y = g(t)

Have continuous derivatives and the parametric curve does not retrace its path for t in [a, b]. The length of the curve is defined by

JUSTIFICATION The infinitesimal piece of the curve (Figure 6.3.4) from t to t+ Δt

is almost a straight line, so its lengthΔs is given by

__________________ (compared to Δt),

__________________ (compared to Δt),

By the Infinite Sum Theorem,

______________________

The general formula for the length of a parametric curve reduces to our first formula when the curve is given by a simple equation x=g(y) or y= f(x).

If y=f(x), a≤ x≤ b, we take x=t and get

____________________

If x=g(y), a≤ y≤ b, we take y=t and get

________________

EXAMPLE 2 Find the length of the path of a ball whose motion is given by

x= 20t, y= 32t - 16t²

from t=0 until the ball hits the ground.(Ground level is y=0, see Figure 6.3.5). The ball is at

ground level when

32t -16t² =0, t=0 and t=2.

We have dx / dt = 20, dy / dt = 32-32t,

We cannot evaluate this integral yet, so the answer is left in the above form. We can get an approximate answer by the Trapezoidal Rule. When x = ____ , the Trapezoidal Approximation is

s ~ 53.5 error ≤ 0.4,

Figure 6.3.5

The following example shows what happens when a parametric curve does retrace its path.

EXAMPLE 3 Let

x = 1 - t², y=1, -1 ≤ t ≤ 1.

As t goes from -1 to 1, the point (x, y) moves from (0,1) to (1,1) and then back along the same line to (0, 1) again. The path is shown in Figure 6.3.6.

The path has length one. However, the point goes along the path twice for a total distance of two. The length formula gives the total distance the point moves.

We next prove a theorem which shows the connection between the length of an arc and the area of a sector of a circle. Given two points P and Q on a circle with center O, the arc PQ is the portion of the circle traced out by a point moving from P to Q in a counterclockwise direction. The sector POQ is the region bounded by the arc PQ and the radii OP and OQ as shown in Figure 6.3.7.

Figure 6.3.7

THEOREM

Let P and Q be two points on a circle with center O. The area A of the sector POQ is equal to one half the radius r times the length s of the arc PQ,

A=____ rs.

DISCUSSION The theorem is intuitively plausible because if we consider an infinitely small arc Δs of the circle as in Figure 6.3.8, then the corresponding sector is almost a triangle of height r and base Δs, so it has area

ΔA ≈ ____ rs. ( compared to Δs ).

DISCUSSION The theorem is intuitively plausible because if we consider an infinitely small arc Δs of the circle as in Figure 6.3.8, then the corresponding sector is almost a triangle of height r and base Δs, so it has area

ΔA ≈ ____ rΔs. (compared to Δs).

Summing up, we expect that A = _____rs.

We can derive the formula C = 2πr for the circumference of a circle using the theorem. By definition, π is the area of a circle of radius one,

_____________________

Then a circle of radius r has area

______________________

Therefore the circumference C is given by

____________________

PROOF OF THEOREM

To simplify notation assume that the center O is at the origin, P is the point

(0, r) on the x-axis, and Q is a point (x, y) which varies along the circle

(Figure 6.3.9). We may take y as the independent variable and

Figure 6.3.9

Use the equation x= __________ for the right half of the circle. Then A

and s depend on y. Our plan is to show that

__________________

First, we find dx/dy:

________________________

Using the definition of arc length,

________________________

The triangle OQR in the figure has area ____xy, so the sector has area

_________________________

Then ________________________________________

Thus _________________________________________

So A and ____ differ and only by a constant. But when y=0, A=___ rs = 0.

Therefore A = ___ rs.

To prove the formula A=___ rs for arcs which are not within a single quadrant we simply cut the arc into four pieces each of which is within a single quadrant.

PROBLEMS FOR SECTION 6.3

Find the lengths of the following curves.

1

3

5

6

8

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

In Problems 26-30, find definite integrals for the lengths of the curves, but do not evaluate the integrals.

26

27

28

29

30

31

32

33

34

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