# How do you find the average value of f(x)=4x^(1/2) as x varies between [0,3]?

Dec 1, 2016

The average value of $f$ on $\left[a , b\right]$ is $\frac{1}{b - a} {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$

#### Explanation:

$\frac{1}{3 - 0} {\int}_{0}^{3} 4 {x}^{\frac{1}{2}} \mathrm{dx} = \frac{4}{3} {\int}_{0}^{3} {x}^{\frac{1}{2}} \mathrm{dx}$

$= \frac{4}{3} \left[{\left.\frac{2}{3} {x}^{\frac{3}{2}}\right]}_{0}^{3}\right]$

$= \frac{4}{3} \left[\frac{2}{3} \left({3}^{\frac{3}{2}} - 0\right)\right]$

$= \frac{4}{3} \left[\frac{2}{3} \left(3 \sqrt{3}\right)\right]$

$= \frac{8 \sqrt{3}}{3}$