NCERT Solutions - Exercise 8.2 : Application of Integrals

 Page 1


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
Exercise 8.2                                                                Page No: 371 
 
1. Find the area of the circle  which is interior to the parabola  
 
Solution:  
Step 1: Equation of the circle is  
 ……….(1) 
 
Here, centre of circle is (0, 0) and radius is 3/2 
Equation of parabola is  …..(2) 
Step 2: To find values of x and y 
Put  in equation (1),  
  
  
  
  
Page 2


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
Exercise 8.2                                                                Page No: 371 
 
1. Find the area of the circle  which is interior to the parabola  
 
Solution:  
Step 1: Equation of the circle is  
 ……….(1) 
 
Here, centre of circle is (0, 0) and radius is 3/2 
Equation of parabola is  …..(2) 
Step 2: To find values of x and y 
Put  in equation (1),  
  
  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
  
  or  
  or  
Find the value of x: 
Put  in , 
  
Put  in , 
  
=>  
Therefore, Points of intersections of circle (1) and parabola (2) are  
A and B 
Step 3: Area OBM = Area between parabola (2) and y-axis 
=  
  
=  
Page 3


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
Exercise 8.2                                                                Page No: 371 
 
1. Find the area of the circle  which is interior to the parabola  
 
Solution:  
Step 1: Equation of the circle is  
 ……….(1) 
 
Here, centre of circle is (0, 0) and radius is 3/2 
Equation of parabola is  …..(2) 
Step 2: To find values of x and y 
Put  in equation (1),  
  
  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
  
  or  
  or  
Find the value of x: 
Put  in , 
  
Put  in , 
  
=>  
Therefore, Points of intersections of circle (1) and parabola (2) are  
A and B 
Step 3: Area OBM = Area between parabola (2) and y-axis 
=  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
  
=  =  
=  ……….(3) 
Step 4: Now area BDM = Area between circle (1) and y-axis 
=  
  
=  
  
=  
=  
Page 4


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
Exercise 8.2                                                                Page No: 371 
 
1. Find the area of the circle  which is interior to the parabola  
 
Solution:  
Step 1: Equation of the circle is  
 ……….(1) 
 
Here, centre of circle is (0, 0) and radius is 3/2 
Equation of parabola is  …..(2) 
Step 2: To find values of x and y 
Put  in equation (1),  
  
  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
  
  or  
  or  
Find the value of x: 
Put  in , 
  
Put  in , 
  
=>  
Therefore, Points of intersections of circle (1) and parabola (2) are  
A and B 
Step 3: Area OBM = Area between parabola (2) and y-axis 
=  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
  
=  =  
=  ……….(3) 
Step 4: Now area BDM = Area between circle (1) and y-axis 
=  
  
=  
  
=  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
=  
=  
=  …….(4) 
Step 5.  
Required shaded area = Area AOBDA = 2 (Area OBD) = 2 (Area OBM + Area MBD) 
=  =  
=  
=  =  
=   
2. Find the area bounded by the curves  and  
Solution:  
Equations of two circles are 
 
 ……….(1) 
 
 
Page 5


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
Exercise 8.2                                                                Page No: 371 
 
1. Find the area of the circle  which is interior to the parabola  
 
Solution:  
Step 1: Equation of the circle is  
 ……….(1) 
 
Here, centre of circle is (0, 0) and radius is 3/2 
Equation of parabola is  …..(2) 
Step 2: To find values of x and y 
Put  in equation (1),  
  
  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
  
  or  
  or  
Find the value of x: 
Put  in , 
  
Put  in , 
  
=>  
Therefore, Points of intersections of circle (1) and parabola (2) are  
A and B 
Step 3: Area OBM = Area between parabola (2) and y-axis 
=  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
  
=  =  
=  ……….(3) 
Step 4: Now area BDM = Area between circle (1) and y-axis 
=  
  
=  
  
=  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
=  
=  
=  …….(4) 
Step 5.  
Required shaded area = Area AOBDA = 2 (Area OBD) = 2 (Area OBM + Area MBD) 
=  =  
=  
=  =  
=   
2. Find the area bounded by the curves  and  
Solution:  
Equations of two circles are 
 
 ……….(1) 
 
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
integrals 
 
And  ……….(2) 
From equation (1),  
Put this value in equation (2), 
 
  
  
  
Put  in , 
   
 The two points of intersections of circles (1) and (2) are  and .