Answer
$\displaystyle \frac{3}{\pi}\ln(2+\sqrt{3})\approx 1.2576$
Work Step by Step
see definition on p.281.
Average value =$\displaystyle \frac{1}{b-a}\int_{a}^{b}f(x)dx$
Using the given data,
Average value =$\displaystyle \frac{1}{2-0}\int_{0}^{2}\sec(\frac{\pi x}{6})dx$
Find the indefinite integral. Use the table on page 333.
$\displaystyle \int\sec(\frac{\pi x}{6})dx=\left[\begin{array}{ll}
u=\frac{\pi}{6}x & \\
du=\frac{\pi}{6}dx & dx=\frac{6}{\pi}du
\end{array}\right]=\displaystyle \frac{6}{\pi}\int\sec udu$
$=\displaystyle \frac{6}{\pi}\ln|\sec u+\tan u|+C$
$=\displaystyle \frac{6}{\pi}\ln|\sec(\frac{\pi}{6}x)+\tan(\frac{\pi}{6}x)|+C$
Average value = $\displaystyle \frac{1}{2}\cdot\frac{6}{\pi}[\ln|\sec(\frac{\pi}{6}x)+\tan(\frac{\pi}{6}x)|]_{0}^{2}$
$=\displaystyle \frac{3}{\pi}[\ln|\sec(\frac{\pi}{3})+\tan(\frac{\pi}{3})|-\ln|\sec 0+\tan 0|]$
$=\displaystyle \frac{3}{\pi}[\ln(2+\sqrt{3})-\ln(1+0)]$
$=\displaystyle \frac{3}{\pi}\ln(2+\sqrt{3})\approx 1.2576$
