Today we get to the Fundamental Theorem of Calculus, which comes in two parts. This theorem is essential, in that it shows how the two seemingly-dissimilar fields of integral and differential calculus are actually two sides of the same coin. From this point, most of the basic theory of integration comes down to finding the “mirror image” of facts about differentiation.
First, let’s start with some continuous function and define a new function by . Now the fundamental theorem tells us that this new function is differentiable, and its derivative is the function we started with! That is:
To see this, let’s consider the difference . The first term here is the integral . Then we can split this interval up to get the sum of the integrals . But the first part here is just , which we’re about to subtract off. Then the difference quotient is . The derivative will be the limit of this difference quotient as goes to .
So now let’s use the Integral Mean Value Theorem to get at the integral here. It tells us that there’s some between and with — the difference quotient exactly! And as gets smaller and smaller, gets squeezed closer and closer to . And because is continuous, we find that . Presto!
A very common metaphor here is to think of a carpet whose width at a point along its length is . Then its total area from the starting point up to is the integral . How fast is the area increasing as we unroll more carpet? As we unroll more length we get more area, and so the derivative of the area is the width of the carpet.