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The mean value theorem

Suppose that the function f(x) is non negative and continuous in the closed interval [a,b]. Then the Riemann integral img 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 img 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)=x2 on the interval [0,2] is 4/3 (remember we have proved that the integral is 8/3).

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 img, such that img. The number μ is called the mean value of f in the interval [a,b]. vai alla risorsa o alla soluzione

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.

mean value theorem

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.

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