## You are here

Homeaverage value of function

## Primary tabs

# average value of function

The set of the values of a real function $f$ defined on an interval $[a,\,b]$ is usually uncountable, and therefore for being able to speak of an *average value* of $f$ in the sense of the average value

$\displaystyle A.V.\;=\;\frac{a_{1}\!+\!a_{2}\!+\ldots+\!a_{n}}{n}\;=\;\frac{% \sum_{{j=1}}^{n}a_{j}}{\sum_{{j=1}}^{n}1}$ | (1) |

of a finite list $a_{1},\,a_{2},\,\ldots,\,a_{n}$ of numbers, one has to replace the sums with integrals. Thus one could define

$A.V.(f)\;:=\;\frac{\int_{a}^{b}\!f(x)\,dx}{\int_{a}^{b}\!1\,dx},$ |

i.e.

$\displaystyle A.V.(f)\;:=\;\frac{1}{b\!-\!a}\int_{a}^{b}\!f(x)\,dx.$ | (2) |

For example, the average value of $x^{2}$ on the interval $[0,\,1]$ is $\frac{1}{3}$ and the average value of $\sin{x}$
on the interval $[0,\,\pi]$ is $\frac{2}{\pi}$.

The definition (2) may be extended to complex functions $f$ on an arc $\gamma$ of a rectifiable curve via the contour integral

$\displaystyle A.V.(f)\;:=\;\frac{1}{l(\gamma)}\int_{\gamma}\!f(z)\,dz$ | (3) |

where $l(\gamma)$ is the length of the arc. If especially $\gamma$ is a closed curve in a simply connected domain where $f$ is analytic, then the average value of $f$ on $\gamma$ is always 0, as the Cauchy integral theorem implies.

## Mathematics Subject Classification

26D15*no label found*11-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Additional Reference

PlanetMath article: non-Newtonian calculus.