Mathematics 162                              Laboratory 12                        Week of April 19, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

The Length of a Curve

In today's laboratory, we will begin developing a method for computing the length of a curve described by a set of parametric equations.

Part 1: Plotting Parameterized Curves

Recall that last week in lab we considered the trajectory of a projectile described by the parametric equations

define the parametric functions,

f[t_] := 800 Cos[Pi/6] t;
g[t_] := -16 t^2 + 800 Sin[Pi/6] t

and plot the trajectory using the command below; select it with the mouse and make it larger; then print out a copy.

ParametricPlot[{f[t], g[t]}, {t,0,25},
                           AspectRatio->Automatic]

One way to estimate the distance traveled by the projectile is to break the trajectory into two (or more) parts and estimate the lengths of those parts.  On the plot you just printed, place a dot at the peak of the trajectory and connect that dot to the initial and terminal points of the trajectory with straight line segments.  The sum of the lengths of those segments will provide a first estimate of the length of the trajectory.  What is that sum?  Show your work and give your answer on your plot.  You may do the calculations either by hand or by using Mathematica.  Have you overestimated or underestimated the actual length of the trajectory?  Explain on the plot.

To get a better estimate of the length of the trajectory, we will break the parameter interval into more subintervals and estimate the distance traveled by the projectile over each of these (shorter) time intervals.  A Mathematica command which can help you visualize this process has been stored in the package Arclength.  Load that package now by typing

<<i:/course/root/math162/arclength.m

Enter the command

segments[{f[t],g[t]}, {t, 0, 25}, 2]

This should produce the picture you drew on your printout.  The idea is that we have broken the time interval 0 ≤ t ≤ 25 into two subintervals, each of length 12.5 seconds

Now use segments with 5 subintervals to see what happens when we break the parameter interval into five equal parts.  Do you think that summing the lengths of these line segments will provide a better estimate of the length of the trajectory?  Explain.

We can use the Sum command to sum the lengths of the line segments in this picture.  Here we use N[Sqrt[...]] rather than Sqrt[...] to speed up the computation.

n = 5; h = (25-0)/n;
Sum[N[ Sqrt[(f[k h] - f[(k-1)h])^2 +
            (g[k h] - g[(k-1)h])^2] ], {k, 1, n}]

Give your estimate of the length of the trajectory, and explain what this command does.

We are going to do more estimates of arc lengths in this lab.  To streamline the work, the group of commands you just used are gathered into one command, estimate, which you already loaded with the package Arclength.  Enter the command below to see that estimate gives the same result as above.

estimate[{f[t],g[t]}, {t,0,25}, 5]

Modify this command to estimate the length of the trajectory using 1-second sub-intervals.  Is your estimate too large or too small?  Use the segments command to obtain visual evidence of the magnitude of the error in this estimate.  Interpret the evidence below.

Now estimate the trajectory length using 0.5-second subintervals.  Is there numerical evidence you have obtained an accurate estimate of the length of the curve?  Explain.

Part 2: Some Curves of Known Length

In this part of the lab, we will test our method by estimating the lengths of some curves whose exact lengths we can compute.  In parts (a)-(c): (1) use ParametricPlot to plot the given curve and (2) compute by hand its exact length;   use the estimate command to estimate the curve's length by breaking the parameter interval into (3) 50 subintervals and (4) 100 subintervals; (5) give the error in each of these estimates

(a) x = t, y = 4t/3, 0 ≤ t ≤ 3.

(b) x = cos t, y = sin t, 0 ≤ t ≤ 2π.

(c)  x = cos t, y = sin t, 0 ≤ t ≤ 6π.

Explain your results in (a); mention how the curves in (b) and (c) are different, and how that explains the difference in the errors you obtained in these two parts.

Part 3:  The Length of an Arch of a Cycloid

Recall from last week that the parametric equations

describe a cycloid, the curve traced out by a distinguished point on a circle of radius 1 rolling along the x-axis.  Estimate the length of one arch of the cycloid with an error no greater than 0.005.  Explain why you think you have achieved the required accuracy.

Part 4:  An Alternative Method

We have estimated the length of a curve by summing the lengths of line segments connecting points on the curve; we might have tried a different approach.  In this part of the lab, you will examine one alternative approach.  Consider a curve given parametrically by

x = x(t), y = y(t), a ≤ t ≤ b.
Divide the parameter interval [a, b] into n subintervals of equal length h = (b – a)/n.  If (x0, y0), ... , (xn, yn) are the points on the curve corresponding to the endpoints of the subintervals, we will estimate the length of the arc from (xk–1, yk–1) to (xk, yk) by
                               |xk – xk–1| + |yk – yk–1|.
Geometrically, this corresponds to summing the lengths of the legs of the right triangle with vertices (xk–1, yk–1), (xk, yk–1), and (xk, yk).

A procedure, legs, in the package Arclength (which you have already loaded) can help you to see what is happening.  For this example we will take as our curve the arc of the parabola y = x2 corresponding to 0 ≤ x ≤ 2.  Define

x[t_] := t; y[t_] := t^2

Run

legs[{x[t],y[t]}, {t, 0, 2}, 4]

to plot the curve and show the triangles obtained by breaking the parameter interval into 4 equal parts.  The idea is to sum the lengths of the blue line segments to approximate the length of the curve.  Now run legs with 10, 20, and 40 subintervals.  By examining these plots, decide whether or not you believe that the procedure described here will do a good job of approximating the length of the curve.  Explain your answer.

Test your answer by actually computing the approximations given by this new technique using:  (i)	two subintervals, 	(ii)	four subintervals.
NOTE: The calculations required here are relatively short and easy.  You can either do them by hand or by using Mathematica.

Explain these results.  What do you conclude about this method for estimating the length of a curve?