**CBSE Class 12 Commerce Mathematics Applications of the Integrals**

CBSE Class 12 Commerce Mathematics Applications of the Integrals:- we provide complete details of CBSE Class 12 Commerce Mathematics

CBSE Class 12 Commerce Mathematics Applications of the IntegralsApplications of the Integrals in this article.

**CBSE Class 12 Commerce Mathematics Applications of the Integrals**

### CBSE Class 12 Commerce Mathematics Applications of the Integrals:-**Introduction**

In geometry, we have learnt formulae to calculate areas

of various geometrical figures including triangles,

rectangles, trapezias and circles. Such formulae are

fundamental in the applications of mathematics to many

real life problems. The formulae of elementary geometry

allow us to calculate areas of many simple figures.

However, they are inadequate for calculating the areas

enclosed by curves. For that we shall need some concepts

of Integral Calculus.

In the previous chapter, we have studied to find the

area bounded by the curve y = f (x), the ordinates x = a,

x = b and x-axis, while calculating definite integral as the

limit of a sum. Here, in this chapter, we shall study a specific

application of integrals to find the area under simple curves,

area between lines and arcs of circles, parabolas and

ellipses (standard forms only). We shall also deal with finding

the area bounded by the above said curves.

### CBSE Class 12 Commerce Mathematics Applications of the Integrals:-**Area under Simple Curves**

In the previous chapter, we have studied

definite integral as the limit of a sum and

how to evaluate definite integral using

Fundamental Theorem of Calculus. Now,

we consider the easy and intuitive way of

finding the area bounded by the curve

y = f (x), x-axis and the ordinates x = a and

x = b. From Fig 8.1, we can think of area

under the curve as composed of large

number of very thin vertical strips. Consider

an arbitrary strip of height y and width dx,

then dA (area of the elementary strip)= ydx,

where, y = f (x).

This area is called the elementary area which is located at an arbitrary position

within the region which is specified by some value of x between a and b. We can think

of the total area A of the region between x-axis, ordinates x = a, x = b and the curve

y = f (x) as the result of adding up the elementary areas of thin strips across the region

PQRSP. Symbolically,

### CBSE Class 12 Commerce Mathematics Applications of the Integrals:-**The area of the region bounded by a curve and a line**

In this subsection, we will find the area of the region bounded by a line and a circle,

a line and a parabola, a line and an ellipse. Equations of above mentioned curves will be

in their standard forms only as the cases in other forms go beyond the scope of this

textbook.

### CBSE Class 12 Commerce Mathematics Applications of the Integrals:-**Area between Two Curves**

Intuitively, true in the sense of Leibnitz, integration is the act of calculating the area by

cutting the region into a large number of small strips of elementary area and then

adding up these elementary areas. Suppose we are given two curves represented by

y = f (x), y = g (x), where f (x) ≥ g(x) in [a, b] as shown in Fig 8.13. Here the points of

intersection of these two curves are given by x = a and x = b obtained by taking

common values of y from the given equation of two curves.

### CBSE Class 12 Commerce Mathematics Applications of the Integrals:-**Historical Note**

The origin of the Integral Calculus goes back to the early period of development

of Mathematics and it is related to the method of exhaustion developed by the

mathematicians of ancient Greece. This method arose in the solution of problems

on calculating areas of plane figures, surface areas and volumes of solid bodies

etc. In this sense, the method of exhaustion can be regarded as an early method

of integration. The greatest development of method of exhaustion in the early

period was obtained in the works of Eudoxus (440 B.C.) and Archimedes

(300 B.C.)

Systematic approach to the theory of Calculus began in the 17th century.

In 1665, Newton began his work on the Calculus described by him as the theory

of fluxions and used his theory in finding the tangent and radius of curvature at

any point on a curve. Newton introduced the basic notion of inverse function

called the anti derivative (indefinite integral) or the inverse method of tangents.

During 1684-86, Leibnitz published an article in the Acta Eruditorum

which he called Calculas summatorius, since it was connected with the summation

of a number of infinitely small areas, whose sum, he indicated by the symbol ‘∫’.

In 1696, he followed a suggestion made by J. Bernoulli and changed this article to

Calculus integrali. This corresponded to Newton’s inverse method of tangents.

Both Newton and Leibnitz adopted quite independent lines of approach which

was radically different. However, respective theories accomplished results that

were practically identical. Leibnitz used the notion of definite integral and what is

quite certain is that he first clearly appreciated tie up between the antiderivative

and the definite integral.

Conclusively, the fundamental concepts and theory of Integral Calculus

and primarily its relationships with Differential Calculus were developed in the

work of P.de Fermat, I. Newton and G. Leibnitz at the end of 17th century.

However, this justification by the concept of limit was only developed in the

works of A.L. Cauchy in the early 19th century. Lastly, it is worth mentioning the

following quotation by Lie Sophie’s:

“It may be said that the conceptions of differential quotient and integral which

in their origin certainly go back to Archimedes were introduced in Science by the

investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis …. The discovery

that differentiation and integration are inverse operations belongs to Newton

and Leibnitz”.

### CBSE Class 12 Commerce Mathematics Applications of the Integrals

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