Mean of A Function

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This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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 of finitely many numbers
is that . In other words, is the *constant* value which when
*added* to itself times equals the result of adding the terms . By analogy, a
defining property of the average value of a function over the interval is that

In other words, is the *constant* value which when *integrated* over equals the result of
integrating over . But the integral of a constant is just

See also the first mean value theorem for integration, which guarantees that if is continuous then there exists a point such that

The point is called the mean value of on . So we write 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

This generalizes the **arithmetic** mean. On the other hand, it is also possible to generalize the **geometric** mean to functions by defining the geometric mean of *f* to be

More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

There is also a *harmonic average* of functions and a *quadratic average* (or *root mean square*) of functions.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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