# What is the average value of a function y=secx tanx on the interval [0,pi/3]?

Jun 5, 2016

$\frac{3}{\pi}$

#### Explanation:

The average value of the function $f \left(x\right)$ on the interval $\left[a , b\right]$ is equal to

$\frac{1}{b - a} {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$

Thus, here, the average value is

$\frac{1}{\frac{\pi}{3} - 0} {\int}_{0}^{\frac{\pi}{3}} \sec x \tan x \mathrm{dx}$

Note that $\frac{d}{\mathrm{dx}} \left(\sec x\right) = \sec x \tan x$, so $\int \sec x \tan x \mathrm{dx} = \sec x + C$.

$= \frac{1}{\frac{\pi}{3}} {\left[\sec x\right]}_{0}^{\frac{\pi}{3}} = \frac{3}{\pi} \left(\sec \left(\frac{\pi}{3}\right) - \sec \left(0\right)\right) = \frac{3}{\pi} \left(2 - 1\right) = \frac{3}{\pi}$