CBSE Class 12 Mathematics Application Of Integration Notes

Download CBSE Class 12 Mathematics Application Of Integration Notes in PDF format. All Revision notes for Class 12 Application Of Integrals have been designed as per the latest syllabus and updated chapters given in your textbook for Application Of Integrals in Standard 12. Our teachers have designed these concept notes for the benefit of Grade 12 students. You should use these chapter wise notes for revision on daily basis. These study notes can also be used for learning each chapter and its important and difficult topics or revision just before your exams to help you get better scores in upcoming examinations, You can also use Printable notes for Class 12 Application Of Integrals for faster revision of difficult topics and get higher rank. After reading these notes also refer to MCQ questions for Class 12 Application Of Integrals given our website

Class 12 Application Of Integrals Revision Notes

Class 12 Application Of Integrals students should refer to the following concepts and notes for Application Of Integrals in standard 12. These exam notes for Grade 12 Application Of Integrals will be very useful for upcoming class tests and examinations and help you to score good marks

Notes Class 12 Application Of Integrals

 

(A) KEY CONCEPTS

1. AREA LYING BELOW THE X-AXIS:

If f(x)≤0 for a≤x≤b,then the graph of y=f(x) lies below x-axis Therefore area bounded by the curve y=f(x),x-axis and the ordinates x=a and x=b is given by

class_12_maths_concept_18

class_12_maths_concept_17

 

2. AREA LYING ABOVE THE X-AXIS:

The area enclosed by the curve y= f(x), x-axis & between the ordinate at x=a & x=b is given

class_12_maths_concept_20

class_12_maths_concept_19

 

3. AREA LYING ON RIGHT OF Y-AXIS :

Area bounded by the curve x=f(y),y-axis and the abscissa y=c and y=d is given by

class_12_maths_concept_23 class_12_maths_concept_22

4. AREA LYING ON LEFT OF Y-AXIS:

The area enclosed by the curve x= f(y), y-axis & between the abscissa at y=c & y=d is given by :

class_12_maths_concept_25

class_12_maths_concept_24

5. AREA BOUNDED BY TWO CURVES

Area bounded by the two curves y = f(x) & y = g(x) where f1(x) f2(x) in a , b & between the ordinate x=a & x=b is given by

class_12_maths_concept_30

 class_12_maths_concept_26

 IMPORTANT FORMULAE TO USE :

class_12_maths_concept_29

 

Important Notes

1. If the equation of the curve contains only even powers of x, then the curve is symmetrical about y-axis

2. If the equation of the curve contains only even powers of y, then the curve is symmetrical about x-axis.

3. If the equation of the curve remains unchanged when x is replaced by x and y by y, then the curve is symmetrical in opposite quadrants.

4. If the equation of the curve remains unchanged when x and y are interchanged ,then the curve is symmetrical about the line y=x

 1. Find the area of the region {(x,y):x2 ≤ y ≤ x }

Sol. The required area is bounded between two curves y =x2 and y= x . Both of these curves are symmetric about y-axis and shaded region in the fig. shows the region whose area is required.

Therefore, required area =2× area of region R1

Now to find point of intersection of curves y =x2 and y= x , we solve them simultaneously.

Clearly, region R1 is in first quadrant, where x>0

x =x => y =x…………….(i)

y =x2…………….(ii)

either x = 0 or x = 1

The limits are , when x=0, y=0 and when x=1, y=1

So points of intersection of the curve are o(0,0) and A(1,1)

Now, required area = 2× area of region R1

CBSE Class 12 Mathematics Application of Integration

CBSE Class 12 Mathematics Application of Integration

CBSE Class 12 Mathematics Application of Integration

CBSE Class 12 Mathematics Application of Integration

CBSE Class 12 Mathematics Application of Integration


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