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       

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:

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

Example 8.6.4   Let $ f(x)=x^2$ , for $ 0 \leq x \leq 2$ .
Figure 4: The length of an arc of parabola.
\begin{figure}\mbox{\epsfig{file=arclength.eps,width=2cm}}\end{figure}
The length of the arc $ OA$ is equal to:

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

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.

Noah Dana-Picard 2007-12-28