Fundamental Theorems

The fundamental theorem of calculus asserts that differentiation and integration are inverse operations, a fact that is not at all obvious, and was not immediately apparent to the inventors of calculus either. The derivative of a function is a measure of the rate of change of the function. On the other hand, the integral of a function from a to b is a measure of the area under the graph of that function between the two points a and b. Specifically, the fundamental theorem of calculus states that if F(x) is a function for which f(x) is the derivative, then the integral of f(x) on the interval [a,b] is equal to F(b) - F(a). The reverse is also true, if F(x) is continuous on the interval [a,b], then the derivative of F(x) is equal to f(x), for all values of x in the interval [a,b]. This theorem lies at the very heart of calculus, because it unites the two essential halves, differential calculus and integral calculus. Moreover, while both differentiation and integration involve the evaluation of limits, the limits involved in integration are much more difficult to manage. Thus, the fundamental theorem of calculus provides a means of finding values for integrals that would otherwise be exceedingly difficult if not impossible to determine.