Arc Length of Curves

(You need Java Runtime Environment to run GraphFunc applet in this website)

 

Apply the definite integral to find the length of a curve, and then use the GraphFunc utility online to confirm the result.  See the demo.

 

Let y = f(x) be defined in an interval [a, b] to represent a smooth curve.  Divide the interval [a, b] into equal small pieces of width.  One of these small pieces may be designated with length, whereby the Pythagorean theorem.  The sum of such line segmentsis the length of a curve.  The curve becomes smooth as the segments and  tend to 0, and we denote these segments in terms of derivative notations, which are dx, dy and ds, respectively.  The Figure 1 shows the arc length (hypotenuse) ds of a small triangle, and it yields the expression:

 

 

Thus, the total length of the curve from the limits x = a and x = b is

 

.

 

Figure 1.

 

Example 1  Find the length of the curvefrom x = 0 to x = 3, and use the GraphFunc utility online to verify the result.

 

Solution

Using yields a length of

  

       (use [1*])

  

                  (I)

 

Use GraphFunc utility online to verify the above result as shown in the following steps:

 

  1. Get into the website at http://graph.seriesmathstudy.com/ (or try a new version at http://graph.seriesmathstudy.com.  Your computer needs JRE 1.5x to run the applet in this website.  If the first time you load this page, you may have to wait for the applet to be loaded.)   Enter the function expression x^2 at command line and click on the Graph It! button to plot the graph.
  2. Enter the limit values x = 0 and x = 3 into the text fields:  .
  3. Click on the Arc Length button to compute the arc length of the curve.   Its result is illustrated in Figure 2.  The computed arc length is marked in red.

 

Figure 2.

 

We see that the arc lengths that GraphFunc computes and the one we derive are the same.

 

 

Example 2  Find the length of the curve y = ln(cos x) from x = 0 to and use the GraphFunc utitlity online to confirm the result.

 

Solution

 

                 

 

 

 

 

 

 ()   

 

We use GraphFunc utility online and follow the steps indicated in the Example 1 to plot the graph and compute the arc length  The result is illustrated in Figure 3.

 

Figure 3.

 

Now you try to use the GraphFunc utility to compute the arc lengths of your graphs online.

 

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