Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. Before the discovery of this theorem, it was not recognized that these two operations were related. The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals. Ĭonversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives.
This implies the existence of antiderivatives for continuous functions. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also known as an indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration.
The two operations are inverses of each other apart from a constant value which is dependent on where one starts to compute area. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating the gradient) with the concept of integrating a function (calculating the area under the curve).
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