Editing Mean value theorem (section)

===First mean value theorem for definite integrals===
[[File:积分中值定理.jpg|thumb|right|Geometrically: interpreting f(c) as the height of a rectangle and ''b''–''a'' as the width, this rectangle has the same area as the region below the curve from ''a'' to ''b''<ref>{{cite web|url=http://www.mathwords.com/m/mean_value_theorem_integrals.htm|title=Mathwords: Mean Value Theorem for Integrals|website=www.mathwords.com}}</ref>]]
Let ''f'' : [''a'', ''b''] → '''R''' be a continuous function. Then there exists ''c'' in (''a'', ''b'') such that

:<math>\int_a^b f(x) \, dx = f(c)(b - a).</math>

This follows at once from the [[fundamental theorem of calculus]], together with the mean value theorem for derivatives.  Since the mean value of ''f'' on [''a'', ''b''] is defined as

:<math>\frac{1}{b-a} \int_a^b f(x) \, dx,</math>

we can interpret the conclusion as ''f'' achieves its mean value at some ''c'' in (''a'', ''b'').<ref name="Comenetz2002">{{cite book|author=Michael Comenetz|title=Calculus: The Elements| year=2002| publisher=World Scientific|isbn=978-981-02-4904-5|page=159}}</ref>

In general, if ''f'' : [''a'', ''b''] → '''R''' is continuous and ''g'' is an integrable function that does not change sign on [''a'', ''b''], then there exists ''c'' in (''a'', ''b'') such that

:<math>\int_a^b f(x) g(x) \, dx = f(c) \int_a^b  g(x) \, dx.</math>