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Suppose that the function *f(x)* is non negative and
continuous in the closed interval [*a,b*]. Then the
Riemann integral exists and represents the
area bounded by the curve *y=f(x)* and the lines
*y=0* (*x*-axis), *x=a* and *x=b*.
Consider a rectangle with bases on the *x*-axis between
*a* and *b*, and with the same area as the
integral. Then its height must represent the
mean value of the function. Of course, the restriction that the
function is non-negative (and also that of being continuous) is
not necessary and can be removed: they are only useful to
simplify geometrical interpretation.

**Example** The average value of the function
*f(x)=x ^{2}* on the interval [

This property is generalized to integrable functions by the following

**Mean value theorem**

Suppose that *f* is integrable in the closed interval
[*a,b*]. Then there exists a number μ, with , such that . The number μ is
called the **mean value** of *f* in the
interval [*a,b*].

If the function *f* is continuous, there exists a number
*c* in the interval [*a,b*] such that
*f(c)=*μ. Unfortunately this theorem gives no hint in
order to find the number μ, otherwise it would be conclusive
as far Riemann integrals are concerned.

From a geometrical point of view this means that every trapezoid
is equivalent to a rectangle: as any rectangle can be
transformed in a square (by rule and compass), it's usual to
say that *every trapezoid can be squared*.

copyright 2000 et seq. maddalena falanga & luciano battaia

first published on january 07 2003 - last updated on september 01
2003