The Fundamental Theorem of Calculus states that the area under the graph of a function over an interval can be calculated by evaluating any antiderivative of the function at the endpoints of the interval. That is, if f is a function defined on an interval [a,b] and F is any antiderivative of f (that is, if F' = f), then the definite integral of f from a to b (i.e. the area under the curve y=f(x) between x=a and x=b) equals F(b) - F(a). Another formulation of this theorem states that the area function of a function f is an antiderivative of f: that is, for any integrable function f, fix a value a in its domain and let Af(x) be the area under the graph of f between a and x. Then Af'(x) = f(x).
What these two equivalent formulations are saying is that, in essence, computing areas and taking derivatives are inverse operations, just as are multiplication and division or addition and subtraction. In the first formulation, if we denote f as F', the theorem states that if we first differentiate F and then calculate the area underneath that derivative, we get back the original function F (up to a constant). The second formulation gives us the reverse statement: if we start by computing the area under the graph of f and then take the derivative of the resulting function, we again get back the original function.
This remarkable theorem is called the Fundamental Theorem of Calculus because it provides the link between the two branches of calculus, differential calculus and integral calculus. Differential calculus is the study of derivatives, which can be thought of as slopes of tangent lines to curves, or as instantaneous rates of change of functions. Integral calculus is the study of areas under curves, which are defined in terms of limits of Riemann sums -- the area under a curve is approximated by a collection of rectangles, and the widths of these rectangles are taken to be smaller and smaller to obtain better and better estimates of the actual area. These two branches of mathematics were developed independently of each other. The insight that they were very closely linked -- for that matter, that they were linked at all -- was due to Sir Isaac Newton and to Gottfried Leibniz, working independently in the late seventeenth century.
From a practical point of view, the theorem gives a simple method for calculating the area under the graph of any function whose antiderivative can be found. For example, suppose we wish to compute the area between 0 and 1 under the curve y=ax. Using Riemann sums, we have to evaluate the limit as n approaches infinity of the sum of ak/n (1/n), where k ranges from 1 to n. However, the Fundamental Theorem tells us that we can simply evaluate any antiderivative of ax at the two endpoints, 0 and 1, to find the area. Since ax/ln(a) is an antiderivative of ax, we can easily find the area to be a1/ln(a) - a0/ln(a) = (a-1)/ln(a).
Conversely, the theorem also gives a method of approximating the antiderivative of a function whose antiderivative cannot be expressed in closed form. For example, the antiderivative of the function f(x) = cos2x cannot be expressed as an algebraic combination of "common" functions (i.e., polynomials, exponentials, logarithms, and trigonometric functions). However, we know from the Fundamental Theorem that the function A(x) that measures the area under the graph of y=cos2t from, say, 0 to the variable endpoint x is an antiderivative of cos2x. Thus for any specific value of x, the value of A(x) can be approximated to any desired level of accuracy by means of Riemann sums.
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