Mean value theorem#Mean value theorems for integration

The mean value theorem of integral calculus (also called Cauchy's mean value theorem ) is an important Theorem of Calculus. It allows one to estimate integrals without calculate the actual value and provides a simple proof of the Fundamental Theorem of Calculus. Here, the Riemann integral is considered. The statement reads:

Let be a continuous function, as well as integrated and either or. Then there exists such that

Applies. Some authors refer to the above statement as an extended mean value theorem and the statement for the mean value theorem or the first mean value theorem. For one gets the important special case:

Which can be interpreted geometrically easy: The area under the curve between and is equal to the contents of a rectangle medium height.

Evidence

Be the interval. The other case can be reduced by the transition to this. Sind and the infimum or the supremum of on, then it follows from the monotonicity and linearity and the Riemann integral:

So there is one with

And it follows from the intermediate value theorem that there is a. Can even show that within the interval can be found.

Condition on g

The condition that applies or is important because otherwise the first inequality would not apply in the proof. In fact, the mean value theorem for functions that do not satisfy this condition holds, in general, not because is for and

And

Second mean value theorem of integral calculus

Be functions, monotonic and continuous. Then there exists such that

In the case that even is continuously differentiable, one can choose. The proof requires partial integration, the Fundamental Theorem of Calculus and the sentence above.