In this video I go over the proof of the mean value theorem for integrals which I covered in my last video. The proof considers a function written as an integral and by applying the original mean value theorem for derivatives the result will yield the mean value theorem for integrals which is very similar to the one for derivatives.
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Mean Value Theorem for Integrals: Proof
The Mean Value Theorem for Integrals:
If f is continuous on [a, b], then there exists a number c in [a, b]such that
The Mean Value Theorem for Integrals is a consequence of the basic or derivative version of the Mean Value Theorem and the Fundamental Theorem of Calculus.
Mean Value Theorem (for Derivatives)
Let f be a function that satisfies the following hypotheses:
- f is continuous on the closed interval [a, b].
- f is differentiable on the open interval (a, b).
Then there is a number c in (a, b) such that:
or, equivalently,
In proofing the Mean Value Theorem for Integrals we can apply the Mean Value Theorem for Derivatives to the following function:
