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 xis the independent variable. The length sis always greater than or equal to 0 because a < band

                          ____________ > 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=tand get

 

                     ____________________

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

                     ________________

 

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

                               x= 20t,   y= 32t - 16t²

 

fromt=0 until the ball hits the ground.(Ground level isy=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 tgoes 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 Pand Qon a circle with center O, the arc PQis the portion of the circle traced out by a point moving from P toQin a counterclockwise direction. The sector POQ is the region bounded by the arc PQand the radii OPandOQas shown in Figure 6.3.7.

 

 

 

 

 

 

 

 

 

Figure 6.3.7

THEOREM

Let Pand Q be two points on a circle with center O. The area Aof the sector POQis equal to one half the radius rtimes the length s of the arc PQ,

 

                         A=____ rs.

 

DISCUSSION   The theorem is intuitively plausible because if we consider an infinitely small arc Δsof the circle as in Figure 6.3.8, then the corresponding sector is almost a triangle of height rand 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 PROBLEMS  FOR  SECTION  6.3

Find the lengths of the following curves.

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

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

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

6    y= _____,    1≤ x ≤ 3             7  x= _____  ,    3≤ y ≤ 6    

8    y= _____,    1≤ x ≤ 100           9  y= _____,    1≤ x ≤ 8     

10   8x=2y4 + y-2,  1≤ y ≤ 2

11   x2/3 + y2/3 =1, first quadrant

12   y=_________dt, 0≤ x ≤ 10

13   y=_________dt, 2≤ x ≤ 6

14   y=_________dt, 1≤ x ≤ 3

15   x=_________dt, 1≤ y ≤ 4

16   y=_________dt, 0≤ x ≤ 1

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

     is x= t3/2

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

     is given by x= 4(1-t)3/2, y= 2t 3/2.

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

     is given by x= t 3/2, y= 9t .

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

     is given by the parametric equations x= 5t2, y= t 3.

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

    is x= cos t, y= sin t .

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

    x= 3cos 2t, y= 3 sin 2t .

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

    x= cos²t, y= sin²t .

24  Find the distance moved by an object with motion

     x= et cos t, y= et sin t . 0≤ t ≤ 1.

25  Let A(t) and L(t) be the area under the curve y=x² from x=0 to x=t, and the

    length of the curve from x=0 to x=t, respectively. Find d(A(t))/d(L(t)).

 

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

26   y=x3,       0≤ x≤1

27   y=2x2 - x + 1, 0≤ x ≤ 4

28   x=1/t, y=t² ,  0≤ t≤5

29   x=2t + 1,    y=____, 1≤ t ≤ 2

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

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

     Trapezoidal Approximation where Δx=___.

 

 

32   Set 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=___.

 

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

     Trapezoidal Approximation where Δx=1

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

     the Trapezoidal Approximation where Δx=_____

 

□35  Suppose 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.

      Assuming all derivatives are continuous and the parametric curve does

      not retrace its path, prove that the two formulas for curve length give the

      same values. Hint: Use integration by change of variables.

 

 

 

 

 

 

 

 

 

 

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

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

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

6    y= _____,    1≤ x ≤ 3             7  x= _____  ,    3≤ y ≤ 6    

8    y= _____,    1≤ x ≤ 100           9  y= _____,    1≤ x ≤ 8     

10   8x=2y4 + y-2,  1≤ y ≤ 2

11   x2/3 + y2/3 =1, first quadrant

12   y=_________dt, 0≤ x ≤ 10

13   y=_________dt, 2≤ x ≤ 6

14   y=_________dt, 1≤ x ≤ 3

15   x=_________dt, 1≤ y ≤ 4

16   y=_________dt, 0≤ x ≤ 1

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

     is x= t3/2

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

     is given by x= 4(1-t)3/2, y= 2t 3/2.

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

     is given by x= t 3/2, y= 9t .

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

     is given by the parametric equations x= 5t2, y= t 3.

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

    is x= cos t, y= sin t .

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

    x= 3cos 2t, y= 3 sin 2t .

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

    x= cos²t, y= sin²t .

24  Find the distance moved by an object with motion

     x= et cos t, y= et sin t . 0≤ t ≤ 1.

25  Let A(t) and L(t) be the area under the curve y=x² from x=0 to x=t, and the

    length of the curve from x=0 to x=t, respectively. Find d(A(t))/d(L(t)).

 

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

26   y=x3,       0≤ x≤1

27   y=2x2 - x + 1, 0≤ x ≤ 4

28   x=1/t, y=t² ,  0≤ t≤5

29   x=2t + 1,    y=____, 1≤ t ≤ 2

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

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

     Trapezoidal Approximation where Δx=___.

 

 

32   Set 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=___.

 

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

     Trapezoidal Approximation where Δx=1

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

     the Trapezoidal Approximation where Δx=_____

 

□35  Suppose 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.

      Assuming all derivatives are continuous and the parametric curve does

      not retrace its path, prove that the two formulas for curve length give the

      same values. Hint: Use integration by change of variables.