# How do you find the average value of x^3 as x varies between 2 and 4?

Feb 7, 2017

$30$

#### Explanation:

The function is continuous in the interval [2 ,4] and the average value from x = 2 to x = 4 is the integral.

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

$= \frac{1}{2} {\left[\frac{1}{4} {x}^{4}\right]}_{2}^{4}$

$= \frac{1}{2} \left[\frac{1}{4} \times {4}^{4} - \frac{1}{4} \times {2}^{4}\right]$

$= \frac{1}{2} \left(64 - 4\right) = 30$