Two Mean Value Theorems
The first one says that if is increasing on and is integrable with respect to , with supremum and infimum in the interval, then there is some “average value” between and . This satisfies
In particular, we should note that if is continuous then the intermediate value theorem tells us that there is some with . That is, there is some such that
When this gives us the old integral mean value theorem back again.
So why does this work? Well, if then both sides are zero and the theorem is trivially true. Now, the lowest lower sum is , while the highest upper sum is . The integral itself, which we’re assuming to exist, lies between these bounds:
So we can divide through by to get the result we seek.
We can get a similar result which focuses on the integrator by using integration by parts. Let’s assume is continuous and is increasing on . Our sufficient conditions tell us that the integral of with respect to exists, and the integration by parts formula says
But the first integral mean value theorem tells us that the integral on the right is equal to for some . Then we can rearrange the above formula to read
So there is some point so that the integral of is the same as the integral of the step function taking the value until and the value after it.