Arc length.

Definition 8.6.1   A function $f$ is smooth on the interval $I$ if the two following conditions are fulfilled:

1. $f$ is differentaible on $I$ ;
2. The first derivative $f'$ is continuous on $I$ .

If $f$ is smooth, its graph is called a smooth curve.

Example 8.6.2       

• The sine function is smooth on $\mathbb{R}$ .
• The absolute value function is smooth on $(0,+\infty)$ , but is mot smooth on $\mathbb{R}$ .

Definition 8.6.3   Let $f$ be a smooth function on the interval $[a,b]$ , where $a<b$ . The length of the graph of $f$ is:

$$L=\int_a^b \sqrt{1+f'(x)^2} \; dx$$

Example 8.6.4   Let $f(x)=x^2$ , for $0 \leq x \leq 2$ .

Figure 4: The length of an arc of parabola.

The length of the arc $OA$ is equal to:

$$L=\int_0^2 \sqrt{1+(2x)^2}\; dx = \left[ \frac 14 \ln \left\vert ... ...rt + \frac 12 x \sqrt{1+4x^2} \right]_0^2=\frac 14 \ln (4+\sqrt{17})+\sqrt{17}.$$

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.