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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 over the interval is defined by
Recall that a defining property of the average value y ¯ {\displaystyle {\bar {y}}} of finitely many numbers y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\dots ,y_{n}} is that n y ¯ = y 1 + y 2 + ⋯ + y n {\displaystyle n{\bar {y}}=y_{1}+y_{2}+\cdots +y_{n}}. In other words, y ¯ {\displaystyle {\bar {y}}} is the constant value which whenadded to itself n {\displaystyle n} times equals the result of adding the n {\displaystyle n} terms y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}}. By analogy, adefining property of the average value f ¯ {\displaystyle {\bar {f}}} of a function over the interval {\displaystyle } is that
In other words, f ¯ {\displaystyle {\bar {f}}} is the constant value which when integrated over {\displaystyle } equals the result ofintegrating f {\displaystyle f} over {\displaystyle }. But the integral of a constant f ¯ {\displaystyle {\bar {f}}} is just
See also the first mean value theorem for integration, which guaranteesthat if f {\displaystyle f} is continuous then there exists a point c ∈ {\displaystyle c\in } such that