Fundamental Theorem Calculus

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions. The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.

The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638 1675). Isaac Barrow (1630 1677) proved the first completely general version of the theorem, while Barrow's student Isaac Newton (1643 - 1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646 1716) systematized the knowledge into a calculus for infinitesimal quantities.

Basic principle of calculus. It relates the derivative to the integral and provides the principal method for evaluating definite integrals. In brief, it states that any function that is continuous over an interval has an antiderivative on that interval. Further, the definite integral of such a function over an interval a < x < b is the difference F(b) - F(a), where F is an antiderivative of the function. This particularly elegant theorem shows the inverse function relationship of the derivative and the integral and serves as the backbone of the physical sciences. It was articulated independently by Isaac Newton and Gottfried Wilhelm Leibniz.

Fundamental Theorem of Calculus.

where F is any antiderivative of f, i.e., F' = f.

Intuitively, the theorem simply says that if you know all the little instantaneous changes in some quantity, then you may compute the overall change in the quantity by adding up all the little changes.

To get a feeling for the statement, we will start with an example. Suppose you travel in a straight line, starting at time t = 0, and with varying speeds. If d(t) denotes the distance from the origin at time t and v(t) denotes the speed at time t, then v(t) is the instantaneous rate of change of d and is therefore the derivative of d. Suppose you know only v(t) from your speedometer, and you want to recover d(t). The fundamental theorem of calculus says that you should integrate v in order to get d. And this is exactly what you would have done, even without knowing that theorem: record the speed at regular intervals, maybe after 1 minute, 2 minutes, 3 minutes and so on, and then multiply the first speed with 1 minute to get an estimate for the distance covered in the first minute, then multiply the second speed with 1 minute to get the distance covered in the second minute etc., and then add all the distances up. In order to get an even better estimate of your current distance, you need to record the speeds at shorter time intervals. The limit as the length of the intervals approaches zero is exactly the definition of the integral of v.