Integrals

Introduction:

(i) Differential Calculus: Differential Calculus is centred on the concept of the derivative. The original motivation for the derivative was the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines.

(ii) Integral Calculus: Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions.

(iii) Anti Derivatives: The functions that could possibly have given function as a derivative are called anti derivatives (or primitive) of the function.

(iv) Integration: The formula that gives all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration.

The development of integral calculus arises out of the efforts of solving the problems of the following types:

(a) the problem of finding a function whenever its derivative is given,
(b) the problem of finding the area bounded by the graph of a function under certain conditions.

The Fundamental Theorem of Calculus:

There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability.

The definite integral is used as a practical tool for:

(i) Science and Ingineering

(ii) Economics, finance and probability.

Integration as an Inverse Process of Differentiation

Integration is the inverse process of differentiation.

Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation.

Exercise 7.1