Okay: time to get back on track. Today, we’ll see a theorem about integrals that’s similar to the Differential Mean Value Theorem. Specifically, it states that if we have a continuous function then there is some so that
Let’s consider the Darboux sums we use to define the integral. We know that if we choose a partition, then its upper Darboux sum is greater than any Riemann sum of any refinement of that partition. So let’s take the absolute coarsest possible partition: the one where we just have partition points and . Then the upper Darboux sum is , where is the maximum value of on the interval . Similarly, the lower Darboux sum on this interval is (where is the minimum value of ), and it’s the lowest possible Darboux sum. Then we can divide everything in sight by to get the inequality
Now the Intermediate Value Theorem tells us that must take every value between and at some point between and . And thus there must exist a so that
just as we wanted.