Question

What is the average value of the function `f(x)=sin4x` on the interval `[-pi,pi]`?

Answer 1

It is `0`.

Explanation:

The average value of an continuous function `f` on interval `[a,b]` is

`1/(b-a) int_a^b f(x) dx`.

Here, we have
`1/(pi-(-pi)) int_(-pi)^pi sin4x dx =1/(2pi) [-1/4 cos(4x)]_-pi^pi`

`= - 1/(8pi) [cos(4pi)-cos(-4pi)] = 1/(8pi) [0] = 0`