Mean value theorem for definite integrals/Riemann/Section

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For a Riemann-integrable function , one may consider

as the mean height of the function, since this value, multiplied with the length of the interval, yields the area below the graph of . The Mean value theorem for definite integrals claims that, for a continuous function, this mean value is in fact obtained by the function somewhere.

Suppose that is a compact interval, and let

be a continuous function. Then there exists some such that

On the compact interval, the function is bounded from above and from below, let and denote the minimum and the maximum of the function. Due to fact, they are both obtained. Then, in particular, for all , and so

Therefore, with some . Due to the Intermediate value theorem, there exists a such that .