ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths

ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions Chapter-3 Section-B. Step by step Solutions of ML Aggarwal ISC Understanding APC Mathematics Class-12 Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions

Class: 12th
Subject: Mathematics
Chapter : Ch-3 Applications of Integrals Section-B
Board ISC Board
Writer ML Aggarwal ISC Understanding ( Vol-II)
 Publications APC Arya Publications ( 2020-21 )

-: Select Topics :-

Exe-3.1, 

Exe-3.2, 

Chapter Test


ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions

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.

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. 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.

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]. 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.

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.


Exercise – 3.1,

ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions

(page-1415 to 1417)

Question 1:

Find the region bounded by ……………………….. and the x-axis (i the first quadrant).

Question 2:

………………….

………………..

…………………..

Question 14:

Determine the area ………………………………. radius r is πr².

Question 15:

……………………..

………………….

……………………

Question 28:

Find the area of the smaller …………………………. 4x.

Question29:

…………………..

…………………….

……………………..

Question 37:

Find the area of enclosed by the curve ……………………..


Exercise – 3.2,

ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions

(page-1420)

Question 1:

Find the area of the region ……………………. x = 4 .

Question 2:

Show that the area ……………..

Question 3:

……………………

…………………

Question 5:

Find the area enclosed by the curve ……………… and the x- axis.


Chapter Test

ML Aggarwal Applications of Integrals ISC Class-12 Understanding APC Maths Solutions

(page-1421)

Question 1:

Find the area under given curve and the given lines :

(i)…..

(ii)…………

Question 2:

Find the area of the region  …………………… y=4

Question 3:

…………………….

…………………….

……………………..

Question 11:

Find the area common to the to the circle ……………. = 4x.

Question 12:

………………..

………………..

Question 15:

Find the ratio in which the area ………………………. by the line  x = 3.

-: End of Applications of Integrals ML Aggarwal ISC Class-12 Solutions :-

Return to :- ML Aggarwal ISC Class-12 APC Understanding Maths Solutions


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