Subject: Calculus

Fundamental Theorem Of Calculus

Diferentiation and integration are two of the most important operations in calculus which are entertwined in many ways. The relationship between this two operations are detailed and specified in a sound manner by the Fundamental Theorem of Calculus. This theorem serves as the "bridge" connecting integration and differentiation.
Basically, the theorem comes in two parts: the First Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus. Both parts tackles on the specification that an integral can be reversed by a differentiation process. Yet, the only difference between the two parts of the fundamental theorem is that the first part deals with indefinite integrals while the second part works on definite integrals.
Moreover, it can also be said that the first part guarantees that antiderivatives of functions exists. The latter part is also succesful in computing definite integral of a function through its antiderivatives thus simplifying the computation of definite integrals.

First Fundamental Theorem of Calculus

Consider a function f which is well-behaved and continuous on the interval [a, b]. Assume that F is function defined at all x in [a, b] such that

F(x)=\int_{a}^{x}f(t)dt.

Then,

F'(x)=f(x)

if and only if F is differentiable and continuous on the interval [a, b].

Second Fundamental Theorem of Calculus (Newton-Leibniz Axiom)

If functions f and g are defined properly on the interval [a, b] and that its derivative and integral exists on [a, b] such that

f(x)=g'(x)

then

\int_{a}^{b}f(x)dx=g(a)-g(b)

NEXT TOPIC: Integration by parts