第6.3节曲线长度

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6.3LENGTH OF A CRUVE

Asegmentof a curve in the plane (Figure 6.3.1) is described by

y=f(x), a≤ x≤ b.

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

DEFINITION

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

y=f(x), a≤x≤b

issmooth.The length of the curve is defined as

Because____________=______________,the equation is sometimes

writtenin the form _________________

withtheunderstanding that xisthe independent variable. The length sisalways greater than or equal to 0 because a < band

____________>0.

JUSTIFICATION Lets (u,w) bethe intuitive length of the curve between t = u andt = w.

Thefunctions(u,w) hasthe Addition Property; the length of the curve from utow equals

thelengthfrom u tov plusthe length from v tow.Figure 6.3.2 shows an infinitesimal

pieceofthe curve from x tox +Δx.Its length is Δs = s (x, x+Δx).

Theslope dy/ dx is a continuous function of x, and therefore changesonly by an infinitesimal amount between x and x+Δy.Hence

Δs≈________ ( compared to Δx).

Dividingby Δx,

Then_________________ (compared to Δx).

Usingthe Infinite Sum Theorem,

EXAMPLE1 Find the length of the curve

y=2x3/2, 0 ≤ x≤ 1

Shownin Figure 6.3.3. We have

dy/dx= 3x1/2,_________________

Putu=1+9x.Then

___________________

Figure6.3.3

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

x=f(t),y=g(t), c ≤ t≤ d.

Anatural example is the path of a moving particle where is time. Wegive a formula for the length of such a curve.

DEFINITION

Supposethefunctions

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

Havecontinuous derivatives and the parametric curve does not retrace itspath for t in [a, b]. The length of the curve is defined by

JUSTIFICATIONThe infinitesimal piece ofthe curve (Figure 6.3.4) from t to t+ Δt

isalmost a straight line, so its lengthΔs is given by

__________________(comparedto Δt),

Bythe Infinite Sum Theorem,

______________________

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

Ify=f(x), a≤x≤b,we take x=tandget

____________________

Ifx=g(y), a≤y≤b,we take y=tandget

________________

EXAMPLE2 Findthe length of the path of a ball whose motion is given by

x=20t, y= 32t - 16t²

fromt=0until the ball hits the ground.(Ground level isy=0,see Figure 6.3.5). The ball is at

groundlevel when

32t-16t² =0,t=0 and t=2.

Wehave dx / dt = 20, dy / dt= 32-32t,

Wecannot evaluate this integral yet, so the answer is left in the aboveform. We can get an approximate answer by the Trapezoidal Rule. Whenx = ____ , the Trapezoidal Approximation is

s~53.5 error ≤0.4,

Figure6.3.5

Thefollowing example shows what happens when a parametric curve doesretrace its path.

EXAMPLE3 Let

x= 1- t², y=1,-1 ≤ t≤ 1.

Astgoesfrom -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.

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

Wenext prove a theorem which shows the connection between the length ofan arc and the area of a sector of a circle. Given two points PandQona circle with center O,the arc PQisthe portion of the circle traced out by a point moving from PtoQina counterclockwise direction. The sector POQisthe region bounded by the arc PQandthe radii OPandOQasshown in Figure 6.3.7.

Figure6.3.7

THEOREM

LetPandQ betwo points on a circle with center O.The area Aofthe sector POQisequal to one half the radius rtimesthe length softhe arc PQ,

A=____rs.

DISCUSSIONThetheorem is intuitively plausible because if we consider an infinitelysmall arc Δsofthe circle as in Figure 6.3.8, then the corresponding sector isalmost a triangle of height randbase Δs,so it has area

ΔA≈____rs.( compared to Δs ).

DISCUSSIONThe theorem is intuitively plausiblebecause if we consider an infinitely small arc Δs of the circle asin Figure 6.3.8, then the corresponding sector is almost a triangleof height r and base Δs, so it has area

ΔA≈____rΔs.(compared to Δs).

Summingup, we expect that A = _____rs.

Wecan derive the formula C = 2πr for the circumference of acircle using the theorem. By definition, π is the area of a circleof radius one,

_____________________

Thena circle of radius r has area

______________________

Thereforethe circumference C is given by

____________________

PROOFOF THEOREM

Tosimplifynotation assume that the center O is at the origin, Pis the point

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

(Figure6.3.9).We may take y as the independent variable and PROBLEMSFOR SECTION 6.3

Findthe lengths of the following curves.

1y= _____(x+2)3/2,0≤ x ≤ 3 2 ·y=(x²+_____)3/2,-2≤ x ≤ 5

3(3y - 1)2= x3, 0≤ x ≤ 24 y= (4/5)x5/4, 0≤ x ≤ 1

5y=(x-1)2/3,1 ≤ x ≤ 9 hint: Solve for x asa function of y.

6y= _____,1≤ x ≤ 37 x= _____ ,3≤ y ≤ 6

8y= _____,1≤ x ≤ 1009 y= _____,1≤ x ≤ 8

108x=2y4 + y-2, 1≤ y ≤ 2

11x2/3 + y2/3 =1,first quadrant

12y=_________dt, 0≤ x ≤10

13y=_________dt, 2≤ x ≤6

14y=_________dt, 1≤ x ≤3

15x=_________dt, 1≤ y ≤4

16y=_________dt, 0≤ x ≤1

17Find the distance travelled from t=0 to t=1 by anobject whose motion

isx=t3/2

18Find the distance moved from t=0 to t=1 by a particlewhose motion

isgivenby x= 4(1-t)3/2,y= 2t 3/2.

19Find the distance travelled from t=1 to t=4 by anobject whose motion

isgivenby x= t 3/2, y= 9t .

20Find the distance travelled from time t=0 to t=3 by aparticle whose motion

isgivenby the parametric equations x= 5t2, y= t 3.

21Find the distance moved from t=0 to t= 2π by an objectwhose motion

isx=cost, y= sint .

22Find the distance moved from t=0 to t= π by an objectwith motion

x=3cos2t,y= 3sin2t.

23Find the distance moved from t=0 to t= 2π by an objectwith motion

x=cos²t, y= sin²t .

24Find the distance moved by an object with motion

x=etcost, y= et sint . 0≤ t ≤ 1.

25Let A(t) and L(t) be the area under the curve y=x²from x=0 to x=t, and the

lengthofthe curve from x=0 to x=t, respectively. Findd(A(t))/d(L(t)).

InProblems 26-30, find definite integrals for the lengths of thecurves, but do not evaluate the integrals.

26y=x3, 0≤ x≤1

27y=2x2 - x + 1, 0≤ x ≤ 4

28x=1/t,y=t² , 0≤ t≤5

29x=2t+ 1, y=____, 1≤t ≤ 2

30The circumference of the ellipse x² + 4y² = 1

31Set up integral for the length of the curve y=____,1≤ x≤ 2 , and find the

TrapezoidalApproximationwhere Δx=___.

32Set up an integral for the length of the curve x= t² -t, y = ____t 3/2,

0≤t ≤ 1 , and find the Trapezoidal Approximation where Δt=___.

33Set up an integral for the length of the curve y= 1/x,1≤ x ≤ 5 , and find the

TrapezoidalApproximationwhere Δx=1

34Set up an integral for the length of the curve y= x²,-1≤ x ≤ 1 , and find

theTrapezoidalApproximation where Δx=_____

□35Suppose the same curve is given in two ways, by a simple equation

y=F(x),a≤x≤b and by parametric equations x= f(t),y=g(t), c≤ t H≤d.

Assumingallderivatives are continuous and the parametric curve does

notretraceits path, prove that the two formulas for curve length give the

samevalues.Hint: Use integration by change of variables.

Figure6.3.9

Usetheequation x= __________ for theright half of the circle. Then A

andsdepend on y. Our plan is to show that

__________________

First,we find dx/dy:

________________________

Usingthe definition of arc length,