San José State University


SIZE="5" COLOR="#FF0000">appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley U.S.A. 

An Extended Mean Value Theorem for Integrals 

Let f(x) be a continuous function on the interval [a, b] and g(x) a nonnegative integrable function over [a, b] such that G=∫_{a}^{b}g(x)dx is positive. Then there exists an x* in (a, b) such that
Proof:
Let m be the greatest lower bound for f(x) on the interval [a, b] and M be the lowest upper bound for f(x) on that same interval. Let G be the value of ∫_{a}^{b}g(x)dx. Then
Since f(x) is a continuous function on [a, b] it attains all values between m and M. This means there is an x* in (a, b) such that
Therefore there exists an x* in (a, b) such that
HOME PAGE OF appletmagic 
