Exercise 8.1 : Application of Integrals NCERT Solutions | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Exercise 8.1                                                                Page No: 365 
 
1. Find the area of the region bounded by the curve  and the lines x=1, x = 4  and 
the x- axis in the first quadrant. 
Solution: Equation of the curve (rightward parabola) is  
 
 
 
  ……….(1) 
Required area is shaded region: 
=  [From equation (1)] 
=   
=   
=  
Page 2


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Exercise 8.1                                                                Page No: 365 
 
1. Find the area of the region bounded by the curve  and the lines x=1, x = 4  and 
the x- axis in the first quadrant. 
Solution: Equation of the curve (rightward parabola) is  
 
 
 
  ……….(1) 
Required area is shaded region: 
=  [From equation (1)] 
=   
=   
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
=  =  =  sq. units 
2. Find the area of the region bounded by  and the x-axis in the first 
quadrant. 
Solution: Equation of the curve (rightward parabola) is  
  ……….(1) 
 
 
Required area is shaded region, which is bounded by curve   and vertical lines x=2, 
x=4 and x-axis in first quadrant. 
=  [From equation (1)] 
=  =  
=  =  
Page 3


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Exercise 8.1                                                                Page No: 365 
 
1. Find the area of the region bounded by the curve  and the lines x=1, x = 4  and 
the x- axis in the first quadrant. 
Solution: Equation of the curve (rightward parabola) is  
 
 
 
  ……….(1) 
Required area is shaded region: 
=  [From equation (1)] 
=   
=   
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
=  =  =  sq. units 
2. Find the area of the region bounded by  and the x-axis in the first 
quadrant. 
Solution: Equation of the curve (rightward parabola) is  
  ……….(1) 
 
 
Required area is shaded region, which is bounded by curve   and vertical lines x=2, 
x=4 and x-axis in first quadrant. 
=  [From equation (1)] 
=  =  
=  =  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
=  =  sq. units 
 
3. Find the area of the region bounded by  and the y- axis in the first 
quadrant. 
Solution: Equation of curve (parabola) is   
or  ……….(1) 
Required region is shaded, that is area bounded by curve  and Horizontal 
lines  and y-axis in first quadrant. 
 
=  
=  =  =  sq. units 
4. Find the area of the region bounded by the ellipse  
Solution: Equation of ellipse is  ……….(1) 
Page 4


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Exercise 8.1                                                                Page No: 365 
 
1. Find the area of the region bounded by the curve  and the lines x=1, x = 4  and 
the x- axis in the first quadrant. 
Solution: Equation of the curve (rightward parabola) is  
 
 
 
  ……….(1) 
Required area is shaded region: 
=  [From equation (1)] 
=   
=   
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
=  =  =  sq. units 
2. Find the area of the region bounded by  and the x-axis in the first 
quadrant. 
Solution: Equation of the curve (rightward parabola) is  
  ……….(1) 
 
 
Required area is shaded region, which is bounded by curve   and vertical lines x=2, 
x=4 and x-axis in first quadrant. 
=  [From equation (1)] 
=  =  
=  =  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
=  =  sq. units 
 
3. Find the area of the region bounded by  and the y- axis in the first 
quadrant. 
Solution: Equation of curve (parabola) is   
or  ……….(1) 
Required region is shaded, that is area bounded by curve  and Horizontal 
lines  and y-axis in first quadrant. 
 
=  
=  =  =  sq. units 
4. Find the area of the region bounded by the ellipse  
Solution: Equation of ellipse is  ……….(1) 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
 
 
 
 
Here  
From equation (1), 
  
 ……….(2) 
for arc of ellipse in first quadrant. 
Ellipse (1) is symmetrical about x-axis and about y-axis (if we change y to –y or x to –x, 
equation remain same). 
Intersections of ellipse (1) with x-axis (y=0) 
Put y=0 in equation (1), we have 
    
Therefore, Intersections of ellipse (1) with x-axis are (0, 4) and (0, -4). 
 
Now again, 
Intersections of ellipse (1) with y-axis (x=0) 
 
Page 5


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Exercise 8.1                                                                Page No: 365 
 
1. Find the area of the region bounded by the curve  and the lines x=1, x = 4  and 
the x- axis in the first quadrant. 
Solution: Equation of the curve (rightward parabola) is  
 
 
 
  ……….(1) 
Required area is shaded region: 
=  [From equation (1)] 
=   
=   
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
=  =  =  sq. units 
2. Find the area of the region bounded by  and the x-axis in the first 
quadrant. 
Solution: Equation of the curve (rightward parabola) is  
  ……….(1) 
 
 
Required area is shaded region, which is bounded by curve   and vertical lines x=2, 
x=4 and x-axis in first quadrant. 
=  [From equation (1)] 
=  =  
=  =  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
=  =  sq. units 
 
3. Find the area of the region bounded by  and the y- axis in the first 
quadrant. 
Solution: Equation of curve (parabola) is   
or  ……….(1) 
Required region is shaded, that is area bounded by curve  and Horizontal 
lines  and y-axis in first quadrant. 
 
=  
=  =  =  sq. units 
4. Find the area of the region bounded by the ellipse  
Solution: Equation of ellipse is  ……….(1) 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
 
 
 
 
Here  
From equation (1), 
  
 ……….(2) 
for arc of ellipse in first quadrant. 
Ellipse (1) is symmetrical about x-axis and about y-axis (if we change y to –y or x to –x, 
equation remain same). 
Intersections of ellipse (1) with x-axis (y=0) 
Put y=0 in equation (1), we have 
    
Therefore, Intersections of ellipse (1) with x-axis are (0, 4) and (0, -4). 
 
Now again, 
Intersections of ellipse (1) with y-axis (x=0) 
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 8-Application of 
Integrals 
 
 
Putting x=0 in equation (1),    
Therefore, Intersections of ellipse (1) with y-axis are (0, 3) and (0,-3). 
 
Now , 
Area of region bounded by ellipse (1) = Total shaded area = 4 x Area OAB of ellipse in first 
quadrant 
=  [ At end B of arc AB of ellipse;  and at end A of arc AB ; ] 
=  =  
=   
=  =  
=  sq. units 
5. Find the area of the region bounded by the ellipse  
Solution: Equation of ellipse is