 Let's do a quick review of section 5.2 from Active Calculus on the second fundamental theorem of calculus. First recall the first fundamental theorem of calculus, which tells us that if f is a continuous function on the integral from a to b and capital F is an antiderivative of little f, then the definite integral from a to b of little f is capital F of b minus capital F of a. The first fundamental theorem in other words tells us that we can compute the exact value of a definite integral of a function by first finding an antiderivative of that function, and then simply finding the difference between the antiderivative evaluated at the upper limit of integration and the antiderivative evaluated at the lower limit of integration. So in addition to being practically important because it tells us a way of computing an integral, the first fundamental theorem is conceptually important because it formalizes the notion that derivatives and integrals are related and in fact they are inverse operations in some sense. An integral can be thought of as an anti-differentiation problem, which is the opposite of differentiating. The second fundamental theorem of calculus looks at this connection from another point of view. It says that if f is a continuous function and c is any constant, then if you look at the function a of x, which is defined to be the integral from c to x of f of t dt, then the derivative of this function is f of x. This is a function that's defined as an integral and the independent variable is x, not t. So as x increases, there is an area that's being swept out between the graph of f and the horizontal axis, and a of x is an area function that measures this area as x changes. One thing the second fundamental theorem says is that the rate at which this area is being accumulated as x changes, that is, that's the derivative of the area function, is precisely f, the function that you're integrating. So here again, derivatives and integrals are seen to be inverses of each other in the sense that the derivative undoes the integral to reveal the function that's inside. Another thing the second fundamental theorem says is that continuous functions on closed intervals always have anti-derivatives because that's what this area function defined as an integral is. It's an anti-derivative for f because it's derivative equals f. So it establishes the existence of anti-derivatives for continuous functions under the right conditions. The first and second fundamental theorems are saying the same thing from different viewpoints, namely that integration and differentiation are inverse processes, the precise nature of that relationship being made rigorous in the theorem statements. So now let's have a look at some examples of the second fundamental theorem in use.