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

Question

9924 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-05T14:06:56

It is $0$ .

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
$1/(pi-(-pi)) int_(-pi)^pi sin4x dx =1/(2pi) [-1/4 cos(4x)]_-pi^pi$

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

What is the average value of the function f(x)=sin4xf(x)=sin4x on the interval [-pi,pi][-pi,pi]?