Theorem on Fundamental Calculus:

Fundamental theorem on calculus gives the relationship between the two main operations of Calculus, namely Differentiation and Integration. It is one of the most profound results in Mathematics. It is so called because it connects the two main themes of Calculus. Integral Calculus is motivated by the problem of finding a function, whenever its derivative is given and the problem of finding the area bounded by the graph of a function under certain conditions. These lead to the two forms of integrals namely indefinite integrals and finite integrals respectively, which together constitute the Integral Calculus. Fundamental theorem of Calculus relates the indefinite integral and the definite integral. There are two parts of the theorem. The first part of the theorem shows that indefinite integration is the reverse process of differentiation.(i.e.) it is used when we are given the derivative and asked to find its primitive(the original function whose derivative is given). This part of the theorem ensures the existence of anti-derivative for continuous functions. The second part of the theorem makes the computation of definite integrals easy, by using one of its infinitely many anti-derivatives. It states that the area under a section of a curve is the difference of the value of the anti-derivative at the upper limit and the value of the anti-derivative at the lower limit. The Fundamental Theorem of Calculus justifies the procedure of evaluating an anti-derivative at the upper and lower limits of integration and taking the difference. Fundamental theorem on Calculus states that if f (x) is integrable on [a,b] and F(x) is an anti-derivative of f(x) on (a,b) which is continuous on[a,b], then

To utilize this theorem we have to find an anti-derivative for the function we are integrating . This theorem does not state which functions does not have anti-derivative. It is important to develop procedures and methods for finding anti-derivative so as to use this theorem completely.