 First of all, welcome to the very last set of videos in this course. You've made it to the end of an entire semester of calculus, which is something that only a very few people on Earth can say that they've done. So stop the video for a moment and just feel good about that. Okay, now let's recap the main ideas from section 4.4 on the all-important fundamental theorem of calculus. Here's where we are so far. We spent a lot of time in this course studying derivatives. Now, we've spent a lot of time studying definite integrals, and we've seen some connections here. The definite integral of f of x from a to b can be interpreted in one way as the total distance traveled by a moving object from x equals a to x equals b. Earlier, we also said that this quantity was the same thing as taking the object's position function, let's call it s, and remind ourselves that we don't know what s is and subtracting s of b minus s of a. But since s is a position function, it's an anti-derivative of the velocity function we're given. So in some way, definite integrals are deeply connected to derivatives by way of the anti-derivative. This connection is made explicit by the fundamental theorem of calculus, which says that if f is a continuous function on the interval from a to b and capital F is any anti-derivative for little f, then the definite integral of little f from a to b is capital F of b minus capital F of a. So think of little f as the velocity function that we know, and capital F as the position function we don't know. What the fundamental theorem of calculus is saying is that the definite integral of little f, which was computable for us only by geometry or by estimation so far, can now be computed exactly by anti-differentiation. The fundamental theorem of calculus provides us a means of computing integrals, in other words, you find an anti-derivative of the integrand and then simply evaluate the endpoints and subtract. That seems really simple except evaluating the integrand can get pretty tricky. However, at least it's something, and if the integrand can be anti-differentiated at all, then the fundamental theorem of calculus gives us a third and this time exact way to find definite integral values. The middle portion of this section will have you work with an activity on some cases where anti-differentiation is relatively easy. Finally, in this section we introduce the main application of the fundamental theorem of calculus and the definite integrals generally the total change theorem. It says that if little f is continuous and differentiable on the interval from a to b, then little f of b minus little f of a, which represents the total change in little f on the interval from a to b, is equal to the definite integral from a to b, not of f but of f prime. Now this is just a generalization of what we already know about velocities and distances. We know that when we integrate a velocity, which is a derivative of something from a to b, we get a total change in position. But now what the total change theorem is saying is that this idea can be applied to any situation where we have a function little f that models some real world quantity that is changing and for which we know something about its derivative.