In this video I go over the Mean Value Theorem for Integrals which states that for any continuous function from x = a to x = b there exists a number c in that interval such that f(c) is equal to the average of the function during that interval. The theorem is a direct consequence of the original Mean Value Theorem which is for derivatives. This theorem also shows that in that interval the average value can be considered the height of a rectangle with the same area as that under the curve.
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Integrals and Average Value: Mean Value Theorem
If T(t) is the temperature at time t, we might wonder if there is a specific time when the temperature is the same as the average temperature. For example, in the temperature function below, there are two such times: one just before noon and one just before midnight.
In general, is there a number c at which the value of a function f is exactly equal to the average value of the function, that is, f(c) = favg?
The following theorem says that this is true for continuous functions.
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. Proof is in next video.
The geometric interpretation of the Mean Value Theorem for Integrals is that, for positive function f, there is a number c such that the rectangle with base [a, b] and height f(c) has the same area as the region under the graph of f from a to b.
My textbook uses the analogy of being able to chop off a mountain at a certain point and using it to fill in the valleys so that the mountain becomes completely flat.