CBSE NOTES for class 12 th
7. Integrals : Mathematics-II class 12 th:English Medium NCERT Book Solutions
NCERT Books Subjects for class 12th Hindi Medium
7. Integrals
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
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See other sub-topics of this chapter:
1. Introduction Mathematics-II class 12
2. Geometrical interpretation of indefinite integral Mathematics-II class 12
3. Methods of Integration Mathematics-II class 12
4. Integration using trigonometric identities Mathematics-II class 12
5. Integrals of Some Particular Functions Mathematics-II class 12
6. Integration by Partial Fractions Mathematics-II class 12
7. Integration by Parts Mathematics-II class 12
8. Integrals of some more types Mathematics-II class 12
9. Definite Integral Mathematics-II class 12
10. Fundamental Theorem of Calculus Mathematics-II class 12
11. Evaluation of Definite Integrals by Substitution Mathematics-II class 12
12. Some Properties of Definite Integrals Mathematics-II class 12
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