# How do you find the average value of f(x)=sqrtx as x varies between [0,4]?

Jan 3, 2017

The average value of $f \left(x\right) = \sqrt{x}$ as $x \in \left[0 , 4\right]$ is $\frac{4}{3}$

#### Explanation:

The integral average of$f \left(x\right)$ over the interval $\left(a , b\right)$ is calculated as:

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

In our case:

$\overline{f} = \frac{1}{4} {\int}_{0}^{4} \sqrt{x} \mathrm{dx} = \frac{1}{4} {\left[\frac{2}{3} {x}^{\frac{3}{2}}\right]}_{0}^{4} = \frac{1}{6} \left(4 \cdot \sqrt{4}\right) = \frac{4}{3}$