NCERT Grade 12-Application of Integrals-Questions

NCERT Solutions for Class 12 Maths

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Application of Integrals

Exercise 1

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

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

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

4.    Find the area of the region bounded by the ellipse 

5.    Find the area of the region bounded by the ellipse 

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

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

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.

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

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

11.   Find the area of the region bounded by the curve y2 = 4x and the line x = 3.

12.  Choose the correct answer:

Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is

13.  Choose the correct answer:

Area of the region bounded by the curve y = 4x, y-axis and the line y = 3 is:

Exercise 2

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

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

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

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

18.  Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

19.  Choose the correct answer:

Smaller area enclosed by the circle x + y = 4 and the line x + y = 2 is:

(A) 2(π – 2)

(B) π – 2

(C) 2π – 1

(D) 2(π + 2)

20. Choose the correct answer:

Area lying between the curves y2 = 4x and y = 2x is:

Exercise 3

21.  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 = 2 and x-axis.

22.  Find the area between the curves y = x and y = x2.

23.  Find the area of the region lying in the first quadrant and bounded by y = 4x4, x = 0, y = 1 and y = 4.

24.  Sketch the graph of y = |x + 3| and evaluate 

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

26.  Find the area enclosed by the parabola y2 = 4ax and the line y = mx.

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

28.  Find the area of the smaller region bounded by the ellipse  and the line 

29.  Find the area of the smaller region bounded by the ellipse  and the line 

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

31. Using the method of integration, find the area enclosed by the curve |x|+|y| = 1.

32.  Find the area bounded by the curve |(x, y): y ≥ x2 and y = |x||.

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

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

35.  Find the area of the region {(x, y): y2 ≤ 4x and 4x2 + 4y2 ≤ 9}.

36.  Choose the correct answer:

Area bounded by the curve y = x3 the x-axis and the ordinate x = -2 and x = 1 is:

37.  Choose the correct answer:

The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by:

38.  Choose the correct answer:

The area of the circle x + y = 16 exterior to the parabola y = 6x.

39.  Choose the correct answer:

The area bounded by the y-axis, y = cos x and y = sin x when  is:

(A) 2(√2 – 1)

(B) √2 – 1

(C) √2 + 1

(D) √2

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