In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain.
One-dimensional In a one-dimensional domain, the mean f ¯ {\displaystyle {\bar {f}}} of a function f(x) over the interval [a, b] is defined by
f ¯ = 1 b − a ∫ a b f ( x ) d x . {\displaystyle {\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.}
This definition can be justified as follows. 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 defined by the property 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 when added 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, a defining property of the average value f ¯ {\displaystyle {\bar {f}}} of a function over the interval [ a , b ] {\displaystyle [a,b]} is that
∫ a b f ¯ d x = ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}{\bar {f}}\,dx=\int _{a}^{b}f(x)\,dx.}
In other words, f ¯ {\displaystyle {\bar {f}}} is the constant value which when integrated over [ a , b ] {\displaystyle [a,b]} equals the result of integrating f ( x ) {\displaystyle f(x)} over [ a , b ] {\displaystyle [a,b]} . But the integral of a constant f ¯ {\displaystyle {\bar {f}}} is just
∫ a b f ¯ d x = f ¯ x | a b = f ¯ b − f ¯ a = ( b − a ) f ¯ . {\displaystyle \int _{a}^{b}{\bar {f}}\,dx={\bar {f}}x{\bigr |}_{a}^{b}={\bar {f}}b-{\bar {f}}a=(b-a){\bar {f}}.}
Mean value theorem
The first mean value theorem for integration guarantees that if f {\displaystyle f} is a continuous function on [ a , b ] {\displaystyle [a,b]} then there exists a point c ∈ ( a , b ) {\displaystyle c\in (a,b)} such that
∫ a b f ( x ) d x = f ( c ) ( b − a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=f(c)(b-a).}
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