# How do you find the average value of f(x)=cosx as x varies between [1,5]?

Nov 23, 2016

$\frac{1}{4} \left(\sin \left(5\right) - \sin \left(1\right)\right) \approx - 0.45010$

#### Explanation:

The average value of the function $f$ on the interval $\left[a , b\right]$ is

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

With the given information this translates into

$\frac{1}{5 - 1} {\int}_{1}^{5} \cos \left(x\right) \mathrm{dx}$

The antiderivative of $\cos \left(x\right)$ is $\sin \left(x\right)$ so

$= \frac{1}{4} {\left[\sin \left(x\right)\right]}_{1}^{5} = \frac{1}{4} \left(\sin \left(5\right) - \sin \left(1\right)\right)$

This is as simplified as we can get without using a calculator.

$\frac{1}{4} \left(\sin \left(5\right) - \sin \left(1\right)\right) \approx - 0.45010$