# What is the average value of the function f(x)=cos(8x) on the interval [0,pi/2]?

Jan 31, 2016

$0$

#### Explanation:

The average value of a function $f$ over the interval $\left(a , b\right)$ is given by $\frac{{\int}_{a}^{b} f \left(x\right) \mathrm{dx}}{b - a}$. For example, average velocity is given by $\frac{{\int}_{0}^{\tau} v \left(t\right) \mathrm{dt}}{\tau}$.

So in this case

$\frac{{\int}_{0}^{\frac{\pi}{2}} f \left(x\right) \mathrm{dx}}{\frac{\pi}{2} - 0} = \frac{{\int}_{0}^{\frac{\pi}{2}} \cos \left(8 x\right) \mathrm{dx}}{\frac{\pi}{2}}$

$= \frac{{\left[\sin \frac{8 x}{8}\right]}_{0}^{\frac{\pi}{2}}}{\frac{\pi}{2}}$

$= \frac{\left[\sin \frac{4 \pi}{8} - \sin \frac{0}{8}\right]}{\frac{\pi}{2}}$

$= \frac{\left[\frac{0}{8} - \frac{0}{8}\right]}{\frac{\pi}{2}}$

$= 0$