Definition 8.6.1
A function
is smooth on the interval
if the two following conditions are fulfilled:
is differentaible on
;
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
.
Figure 4:
The length of an arc of parabola.
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.