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) represent a smooth curve on the interval [a, b], as shown in Figure 1.  The arc length S of the curve f(x) between x = a and x = b is , . Figure 1.

Example 1  Find the arc length of the curve from 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. Begin by going to http://graph.seriesmathstudy.com (or try a new version at http://newgraph.seriesmathstudy.com.  Your computer needs JRE 1.5x to run the applet in this website.  If this is 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 arc 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.

[1*]  