Finding the average of the absolute value of a function?

Question

I know that to find the absolute value of a function, $$\frac{1}{b-a}\int^b_af(x)\mathrm{d}x$$.

This is actually kind of intuitive. The problem is I don't know how to to find the average of the absolute value of a function. This would be fun to work out on my own, but I might use it on a test in the imminent future, I'd like to know.

I tried a few intuitive exploratory methods, such as splitting $f(x)$ into sections based on wether or not it was below or above $0$, then adding up their absolute values and diving by the number of segments.. didn't seem to work though.

This is to compute the speed (avg. of $|f(x)|$) and velocity (avg. of $f(x)$).

Any help is appreciated!

Answer 1

Have you seen the breakdown of $f(x)$ into $f^+(x)$ and $f^-(x)$? The definitions are $f^+(x) = \begin{cases}f(x) & f(x)\ge 0 \\ 0 & \text{otherwise}\end{cases}$ and $f^-(x) = \begin{cases}f(x) & f(x)\le 0 \\ 0 & \text{otherwise}\end{cases}$. So, $|f(x)| = f^+(x)-f^-(x)$.

Then, the average value would be:

$\displaystyle \dfrac{1}{b-a}\int_a^b|f(x)|dx = \dfrac{1}{b-a}\int_a^b(f^+(x)-f^-(x))dx = \dfrac{1}{b-a}\left(\int_a^b f^+(x)dx - \int_a^b f^-(x)dx\right)$

This is essentially the same thing as splitting up the integral depending on when $f(x)$ is positive/negative, but a little more straightforward to describe notationally. Perhaps this can help you visualize the solution.

Answer 2

Recall: $$|f(x)| = \begin{cases} f(x), & f(x) \geq 0 \\ -f(x), & f(x) < 0\text{.} \end{cases}$$

Suppose in $[a,b]$ that the set of intervals where $f(x) \geq 0$ is $F^{+}$ and the set of intervals where $f(x) < 0$ is $F^{-}$. Then the average value of $|f|$ in $[a,b]$ is $$\dfrac{\int\limits_{F^+}f(x)\text{ d}x + \int\limits_{F^{-}}-f(x)\text{ d}x}{b-a}\text{.}$$