San José State University
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An Extended Mean Value Theorem for Integrals |
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Let f(x) be a continuous function on the interval [a, b] and g(x) a non-negative integrable function over [a, b] such that G=∫abg(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 ∫abg(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
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