And now we come to the second part of the FToC. This takes the first part and flips it around.
We again start with a continuous function , but now we take any antiderivative , so that . Then the FToC asserts that
Before we differentiated a function we got by integrating to get back where we started. Now we’re integrating a function we get by differentiating, and again get back where we started. Integration and differentiation are two sides of the same coin.
Let’s consider a partition of with points . Then we see that . We can add and subtract the value of at each of the intermediate points to see that
Now the Differential Mean Value Theorem tells us that there’s a point so that . And we assumed that , so we have
But this is a Riemann sum for the partition we chose, using the points as the tags. Since every partition, no matter how fine, has such a Riemann sum, the integral must take this value, and the second part of the FToC holds.