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 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: 
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.
()
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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[1*]
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