Average Function Value — Question 8

Question 8

Find the average value of the function $$f(x)=|x|$$ on the interval $[-2,2]$.

Question 8 - Solution

The average value of a function on $[a,b]$ is $$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

Here $a=-2$ and $b=2$, so $$f_{\text{avg}}=\frac{1}{4}\int_{-2}^{2}|x|\,dx.$$

Since $|x|$ is an even function, $$\int_{-2}^{2}|x|\,dx = 2\int_{0}^{2}x\,dx.$$

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

Compute the average value: $$f_{\text{avg}}=\frac{1}{4}\cdot 4=1.$$

$$\boxed{1}$$