Mean of a function

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 one variable, 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 to itself ${\displaystyle n}$ times equals the result of adding the ${\displaystyle n}$ terms of ${\displaystyle y_{i}}$. By analogy, a defining property of the average value ${\displaystyle {\bar {f}}}$ of a function over the interval ${\displaystyle [a,b]}$ is that

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