integral mean value theorem

The Integral Mean Value Theorem.

If f and g are continuousMathworldPlanetmath real functions on an interval [a,b], and g is additionally non-negative on (a,b), then there exists a ζ(a,b) such that


Since f is continuous on a closed bounded set, f is bounded and attains its bounds, say f(x0)f(x)f(x1) for all x[a,b]. Thus, since g is non-negative for all x[a,b]


Integrating both sides gives


If abg(x)dx=0, then g(x) is identically zero, and the result follows trivially. Otherwise,


and the result follows from the intermediate value theorem. ∎

Title integral mean value theorem
Canonical name IntegralMeanValueTheorem
Date of creation 2013-03-22 17:15:56
Last modified on 2013-03-22 17:15:56
Owner me_and (17092)
Last modified by me_and (17092)
Numerical id 9
Author me_and (17092)
Entry type Theorem
Classification msc 26A06
Related topic EstimatingTheoremOfContourIntegral