Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 298: 42

$\frac{4 \sqrt{3}}{\pi}$

The average is given by $$ \frac{1}{\pi / 3-\pi / 6} \int_{\pi / 6}^{\pi / 3} \sec ^{2} x d x=\frac{6}{\pi} \int_{\pi / 6}^{\pi / 3} \sec ^{2} x d x\\=\left.\frac{6}{\pi} \tan x\right|_{\pi / 6} ^{\pi / 3}=\frac{6}{\pi}\left(\sqrt{3}-\frac{\sqrt{3}}{3}\right)=\frac{4 \sqrt{3}}{\pi} $$