Average Value of a Function

The mean of f over an interval, and where it is attained.

The idea

The average value of a function extends the ordinary average — add the values, divide by their count — to a quantity that varies continuously. A function on $[a, b]$ takes a value at every one of infinitely many points, so there is no count to divide by.

Instead, sample $f$ at $n$ evenly spaced points with spacing $\Delta x = \frac{b - a}{n}$ and average the readings. Dividing by $n$ is the same as multiplying by $\frac{\Delta x}{b - a}$, so this average is $\frac{1}{b - a} \sum f(x_i) \, \Delta x$. Finer sampling turns the sum into an integral, which gives the definition.

Definition (Average value).

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

Geometrically, $f_{\text{avg}}$ is the height of the rectangle on $[a, b]$ whose area equals the area under $f$.

A continuous $f$ attains this value somewhere.

Theorem (Mean Value Theorem for Integrals).

If $f$ is continuous on $[a, b]$, then there is a number $c$ in $[a, b]$ with $f(c) = f_{\text{avg}}$, that is, $\int_a^b f(x)\,dx = f(c)\,(b - a)$.

Ways to work on it

Not sure where to start? Take the ten-question placement test.