In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the ”average" value of the function over its domain.

In a one-dimensional domain, the mean of a function f(x) over the interval (a,b) is defined by:

Recall that a defining property of the average value ${\displaystyle {\bar {y}}}$ of finitely many numbers ${\displaystyle y_{1},y_{2},\dots ,y_{n}}$ is that ${\displaystyle n{\bar {y}}=y_{1}+y_{2}+\cdots +y_{n}}$. In other words, ${\displaystyle {\bar {y}}}$ is the constant value which when added ${\displaystyle n}$ times equals the result of adding the ${\displaystyle n}$ terms ${\displaystyle y_{1},\dots ,y_{n}}$. By analogy, a defining property of the average value ${\displaystyle {\bar {f}}}$ of a function over the interval ${\displaystyle [a,b]}$ is that

In other words, ${\displaystyle {\bar {f}}}$ is the constant value which when integrated over ${\displaystyle [a,b]}$ equals the result of integrating ${\displaystyle f(x)}$ over ${\displaystyle [a,b]}$. But the integral of a constant ${\displaystyle {\bar {f}}}$ is just

See also the first mean value theorem for integration, which guarantees that if ${\displaystyle f}$ is continuous then there exists a point ${\displaystyle c\in (a,b)}$ such that

The point ${\displaystyle f(c)}$ is called the mean value of ${\displaystyle f(x)}$ on ${\displaystyle [a,b]}$. So we write ${\displaystyle {\bar {f}}=f(c)}$ and rearrange the preceding equation to get the above definition.

In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

where ${\displaystyle {\hbox{Vol}}(U)}$ and ${\displaystyle dV}$ are, respectively, the domain volume and volume element (or generalizations thereof, e.g., volume form).