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Class 12 Maths NCERT Solutions For Chapter 8 Application of Integrals

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Chapter 8 Application of Integrals

Download NCERT Solutions for Class 12 Mathematics

(Link of Pdf file is given below at the end of the Questions List)

In this pdf file you can see answers of following Questions

EXERCISE 8.1


Question 1. Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.


Question 2. Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. Fig


Question 3. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.


Question 4. Find the area of the region bounded by the ellipse 2 2 1 16 9 x y + = .


Question 5. Find the area of the region bounded by the ellipse 2 2 1 4 9 x y + = .


Question 6. Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y and the circle x2 + y2 = 4.


Question 7. Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line 2 x= a .


Question 8. The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.


Question 9. Find the area of the region bounded by the parabola y = x2 and y = x .


Question 10. Find the area bounded by the curve x2 = 4y and the line x = 4y – 2.


Question 11. Find the area of the region bounded by the curve y2 = 4x and the line x = 3. Choose the correct answer in the following Exercises 12 and 13.


Question 12. Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is
(A) π
(B) 2 π
(C) 3 π
(D) 4 π


Question 13. Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is
(A) 2
(B) 9 4
(C) 9 3
(D) 9 2


EXERCISE 8.2


Question 1. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.


Question 2. Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y2 = 1.


Question 3. Find the area of the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 3 .


Question 4. Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).


Question 5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.Choose the correct answer in the following exercises 6 and 7.


Question 6. Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is
(A) 2 (π – 2)
(B) π – 2
(C) 2π – 1
(D) 2 (π + 2)


Question 7. Area lying between the curves y2 = 4x and y = 2x is
(A) 2 3
(B) 1 3
(C) 1 4
(D) 3 4


Miscellaneous Exercise on Chapter 8


Question 1. Find the area under the given curves and given lines:
(i) y = x2, x = 1, x = 2 and x-axis
(ii) y = x4, x = 1, x = 5 and x-axis


Question 2. Find the area between the curves y = x and y = x2.


Question 3. Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4.


Question 4. Sketch the graph of y = x + 3 and evaluate 0 6 3 − ∫ x + dx .


Question 5. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.


Question 6. Find the area enclosed between the parabola y2 = 4ax and the line y = mx.


Question 7. Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12.


Question 8. Find the area of the smaller region bounded by the ellipse 2 2 1 9 4 x + y = and the line 1 3 2 x y + = .


Question 9. Find the area of the smaller region bounded by the ellipse 2 2 2 2 x y 1 a b + = and the line 1 x y a b + = .


Question 10. Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and the x-axis.


Question 11. Using the method of integration find the area bounded by the curve x + y = 1 . [Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1 and – x – y = 1].


Question 12. Find the area bounded by curves {(x, y) : y ≥ x2 and y = | x |}.


Question 13. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B (4, 5) and C (6, 3).


Question 14. Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0


Question 15. Find the area of the region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9} Choose the correct answer in the following Exercises from 16 to 20. 16. Area bounded by the curve y = x3, the x-axis and the ordinates x = – 2 and x = 1 is
(A) – 9
(B) 15 4 −
(C) 15 4
(D) 17 4


Question 17. The area bounded by the curve y = x | x | , x-axis and the ordinates x = – 1 and x = 1 is given by (
A) 0
(B) 1 3
(C) 2 3
(D) 4 3
[Hint : y = x2 if x > 0 and y = – x2 if x < 0].


Question 18. The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is
(A) 4 (4 3) 3 π −
(B) 4 (4 3) 3 π +
(C) 4 (8 3) 3 π −
(D) 4 (8 3) 3 π +


Question 19. The area bounded by the y-axis, y = cos x and y = sin x when 0 2 x π ≤ ≤ is
(A) 2 ( 2 −1)
(B) 2 −1
(C) 2 +1
(D) 2


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