In the previous video, we introduced and applied the Fundamental Theorem of Calculus, which provides an elegant formula, under certain conditions, for finding exact values for definite integrals, that is, areas under curves. In today's video, we explain the connection between areas and derivatives, which is the main idea that leads to a proof of this remarkable theorem. We also apply the main idea to use the derivative to relate areas of circles to perimeters, and volumes of spheres to surface areas. Recall that the definite integral is the area under the curve y equals f(x) as x goes from a, the left-hand endpoint of the interval, to b, the right-hand endpoint. Assuming the curve is continuous, then the area is described succinctly by the Fundamental Theorem of Calculus, as capital F of b minus capital F of a, where capital F of x is any antiderivative of little f of x, by which we mean that its derivative is little f of x. The main purpose of today's video is to explain why this works. Before starting, it should be remarked that this is one of the most important theorems in mathematics. Though the Ancient Greeks employed thought experiments with approximations and limits, the basic ingredients of calculus, it took a couple of thousand years before mathematicians were able to recognize and articulate the main ideas that lead to the statement and proof of the theorem. So you shouldn't expect the ideas to be straightforward. We have to be radical in our thinking, flexible and prepared to experiment and abruptly change our point of view. The first step is to change the name of the input variable that is used to form the curve with respect to which we're taking the area. Let's use t instead of x, because we want to reserve x for something else in a moment. Wherever we see an x, we replace it with a t. So the horizontal axis is labeled by t. The curve is a function of t, and the integrand and differential use t. We're still finding the same area under the same curve, just using a different name for the input variable. What's so radical about that? We do something completely unexpected. We still allow x to move from a to b, but now use x instead of b as the right-hand endpoint of an interval, with respect to which we take areas under the curve. We create what we call an area function, capital A of x, using the same curve, but now representing the area under the curve from a up to x, wherever x happens to be in the interval. The value of A of x clearly is able to vary as x varies. Being an area under the curve, the value A of x is still represented by a definite integral, but now x appears as the upper terminal, instead of having a role in the specification of the integrand and differential. On the face of it, this looks like a ridiculous and unnecessary complication, as we're only interested in the area under the curve from a to b. But you've already seen many examples in earlier videos, where we've made things more complicated in order to get past a barrier or obstruction, to solve a problem or make things simpler later. The phenomenon you're about to witness is one of the most spectacular examples of problem solving in the history of mathematics. Calculus is a study of how things change, especially about instantaneous rates of change. Let's think about the way the area function changes. First of all, make note of what happens at the two extremes, when x equals a and x equals b, the endpoints of the interval. When x equals little a, both terminals in the definite integral are the same and remember this gives an area of zero. In the diagram below, when x equals a, the green area vanishes. However, if x equals b, then A of b becomes our original definite integral, and the green area under the curve goes all the way from a to b. What about the instantaneous rate of change of the area function? This is the derivative, defined as the limit as h goes to zero of A of x plus h minus A of x over h. We relate the ingredients of this limit to the diagram representing the area function. We assume throughout the following discussion that h is small and positive, and that the curve y equals f of t is increasing as in this diagram. These assumptions greatly simplify the argument, but nevertheless suffice to illustrate all of the important key ideas. Because h is positive, x plus h is slightly to the right of x. We move up to the curve, and note f of x and f of x plus h on the y-axis. There's a small piece of the curve over the interval from x to x plus h, which we enclose with lower and upper rectangles. The height of the lower rectangle is f of x, and the height of the upper rectangle is f of x plus h. The width of both rectangles is h. We color in the area of the lower rectangle blue and the area under the curve between the lower and upper rectangles beige. The value of A of x plus h is the area all the way from a to x plus h, whilst the value of A of x, is the area from a to x. So when you take A of x away from A of x plus h, you just get the area under the curve between x and x plus h, which is just the blue area plus the beige area. We color, in pink, the area in the upper rectangle that's not under the curve. So, if we add this to the blue and beige areas, then we get an upper bound for A of x plus h minus A of x. On the other hand, if we only consider the blue area, then we also get a lower bound. But the blue area is just the area of the lower rectangle, which is h times f of x, and the blue, beige and pink areas combine to give the area of the upper rectangle which is h times f of x plus h. So we get this nice cascade of inequalities, with A of x plus h minus A of x sandwiched in the middle. Let's divide through by the positive number h, retaining the same inequalities throughout, so that the h's cancel on the left and right, and we conclude that f of x is less than or equal to A of x plus h minus A of x over h, which is less than or equal to f of x plus h. So let's see what happens as h goes to zero. f of x just stays the same, and f of x plus h becomes f of x, because we're assuming throughout that the function f is continuous. The expression in the middle goes to the derivative A dashed of x. But this must become f of x, by the Squeeze Law, since both the expressions on the left and right have f of x as the same limit. There's nowhere else to move, but for all three limits to coincide with f of x. Reflect on what we've just done. We defined an area function by allowing the upper terminal of the definite integral, which we now call x, to vary from a to b, and discovered that its derivative is simply the value of f at x. How remarkable? We've learnt so many rules for manipulating derivatives, and we're used to analyzing and creating complicated functions, and the area function looks like it probably is some very weird and exotic function. But when we differentiate it with respect to x, we just reproduce the value of f at x. This beautiful, simple and elegant relationship between areas and curves evaded detection for thousands of years. What's this got to do with the Fundamental Theorem of Calculus? Well, it tells us that the area function is an antiderivative of the function that gives rise to the original curve. The Fundamental Theorem gives us a formula in terms of capital F of x, which is an antiderivative of little f of x. But remember, antiderivatives of a given function differ by a constant. So capital F of x must equal A of x plus C for some constant C. To find C, we can sub in x equals a, so F of a becomes capital A of little a plus C, which is the definite integral with a as both upper and lower terminals, plus C, which is zero plus C, which of course is just C. So the constant is F of a. So F of x becomes A of x plus F of a. In particular, F of b becomes A of b plus F of a. And rearranging gives A of b equal to F of b minus F of a. But A of b is the original area under the curve over the entire interval from a to b, which shows that our original definite integral is indeed given by the formula provided by the Fundamental Theorem of Calculus, finally completing the proof. Now, there were certain simplifications in the way we set up this proof in the accompanying diagrams, so it's really only a sketch. But all of the main ideas behind the Fundamental Theorem have been captured by what we've done.