# Arc length.

Definition 8.6.1   A function is smooth on the interval if the two following conditions are fulfilled:
1. is differentaible on ;
2. The first derivative is continuous on .
If is smooth, its graph is called a smooth curve.

Example 8.6.2

• The sine function is smooth on .
• The absolute value function is smooth on , but is mot smooth on .

Definition 8.6.3   Let be a smooth function on the interval , where . The length of the graph of is:

Example 8.6.4   Let , for .
The length of the arc is equal to:

Remark: To compute this integral, you can either use a table of integrals (for example http://torte.cs.berkeley.edu:90/tilu), or make an appropriate substitution. The example 5.20 can help you too.

Noah Dana-Picard 2007-12-28