CBSE Class 12 — Applications of Integrals
100 Super Revision Test
Are you gearing up for your CBSE Class 12 Maths exam and feeling a bit unsure about Applications of Integrals? Don’t worry — this 100 Super Revision Test is designed exactly for that!
This chapter is one of the most important and scoring parts of calculus. It helps you understand how integration is used to find areas under curves, between curves, and bounded regions — concepts that often appear in board exams.
In this post, you’ll find 100 carefully crafted questions covering all difficulty levels — from simple to challenging. Each question comes with a clear, solution so you can quickly revise, check your understanding, and build confidence before the exam.
Whether you’re revising at the last minute or preparing systematically, this test will help you:
- Strengthen your conceptual clarity
- Practice a wide range of question types
- Get a feel of real exam-style problems
So grab your notebook, a calculator, and let’s dive deep into mastering Applications of Integrals — one question at a time! π
Questions With Answers
1. Area under y = x² from x = 0 to x = 3
9
2. Area bounded by y = √x, x-axis, x = 0 to x = 4
16/3
3. Area enclosed by y = x and y = x²
1/6
4. Area between y = sin x and y = cos x from x = 0 to x = Ο/2
2(√2 − 1)
5. Area enclosed by y = 1 − x² and y = 0
4/3
6. Volume of solid by revolving y = √x, 0 ≤ x ≤ 4, about the x-axis
8Ο
7. Volume (shell) of area under y = x, 0 ≤ x ≤ 2, revolved about the y-axis
16Ο/3
8. Volume when area between y = x² and y = 4 is revolved about the x-axis
256Ο/5
9. Area under y = x³ from x = 0 to x = 2
4
10. Area between y = x² and x = y² in the first quadrant
1/3
11. Pappus' theorem (statement)
V = 2Ο × (area × centroid y-coordinate)
12. Area bounded by y = 2x + 1 and y = x²
Integral form: ∫_{1−√2}^{1+√2} (2x + 1 − x²) dx
13. Area enclosed by y = 4 − x² and y = 2
8√2 / 3
14. Volume when area under y = √(1 − x²) from x = −1 to 1 revolved about x-axis
4Ο/3
15. Area under y = ln x from x = 1 to x = e
1
16. Shell volume of region y = x², x ∈ [0,1], about y-axis
Ο/2
17. Evaluate ∫_0^1 1/(1 + x²) dx
Ο/4
18. Area between y = 2x and y = x² + 1
0 (they touch at x = 1)
19. Area under y = cos x from 0 to Ο (absolute area)
2
20. Area under y = 1/x from x = 1 to x = 4
ln 4
21. Area under y = x³ − 2x from x = 0 to x = 2
0
22. Area bounded by y = x² and y-axis from x = 0 to x = 3
9
23. Area between y = 1/x and x-axis from x = 1 to 5
ln 5
24. Volume of solid generated by revolving y = x from x = 0 to 3 about x-axis
9Ο
25. Area between y = x² + 1 and y = 3x − 2
Integral form: ∫ ( (3x − 2) − (x² + 1) ) dx over intersection interval
26. Area under y = sin x from 0 to Ο
2
27. Area between y = e^x and y-axis from x = 0 to 1
e − 1
28. Volume of solid formed by revolving y = x² from x = 0 to 1 about x-axis
Ο/5
29. Area under y = x⁴ from x = 0 to 2
32/5
30. Area between y = x and y = 2x − 1
1/2
31. Area under y = √(1 − x²) from x = 0 to 1
Ο/4
32. Volume of solid when y = x³, x = 0 to 2, revolved about x-axis
128Ο/7
33. Area between y = x² + 1 and y = 2x + 3
4√3
34. Area under y = 1/x² from x = 1 to 4
3/4
35. Volume of solid formed by revolving y = √x from x = 0 to 9 about x-axis
81Ο/2
36. Area between y = sin x and y = cos x from x = 0 to Ο/4
√2 − 1
37. Area under y = ln x from x = 1 to 4
4 ln 4 − 3
38. Volume of solid by revolving y = x² + 1, x = 0 to 1, about x-axis
28Ο/15
39. Area under y = e^x from x = 0 to 2
e² − 1
40. Area between y = x² and y = 4x − 4
0 (the curves are tangent at x = 2; no finite enclosed region)
41. Area under y = cos x from x = 0 to Ο/3
√3 / 2
42. Volume of solid when y = 1/x from x = 1 to 2 revolved about x-axis
Ο/2
43. Area under y = 1/(1 + x²) from x = 0 to 1
Ο/4
44. Area between y = x² and y = 2x from x = 0 to 2
4/3
45. Volume when area under y = √(4 − x²) from x = −2 to 2 revolved about x-axis
32Ο/3
46. Area under y = x³ from x = 1 to 3
20
47. Volume of solid formed by revolving y = x, x = 0 to 4, about x-axis
64Ο/3
48. Area under y = 1/(1 + x) from x = 1 to 3
ln 2
49. Area between y = x² and y = x + 2
9/2
50. Area under y = sin² x from x = 0 to Ο
Ο/2
51. Area under y = x³ from x = 0 to 1
1/4
52. Volume of solid when y = x² from x = 0 to 2 revolved about x-axis
32Ο/5
53. Area under y = √x from x = 0 to 9
18
54. Area between y = x² and y = 2x from x = 0 to 2
4/3
55. Volume of solid when y = sin x from x = 0 to Ο/2 revolved about x-axis
Ο²/4
56. Area under y = cos x from 0 to Ο/2
1
57. Area under y = 1/x² from x = 1 to 2
1/2
58. Volume when y = e^x from x = 0 to 1 revolved about x-axis
(Ο/2)(e² − 1)
59. Area under y = ln x from x = 1 to e
1
60. Area between y = x² and y = x + 2
9/2
61. Area under y = x⁴ from x = 0 to 1
1/5
62. Volume when y = √x from x = 0 to 4 revolved about x-axis
8Ο
63. Area under y = x² + 1 from x = 0 to 1
4/3
64. Area under y = 1/(1 + x²) from x = 0 to 1
Ο/4
65. Volume when y = x³ from x = 0 to 1 revolved about x-axis
Ο/7
66. Area under y = x² from x = 0 to 2
8/3
67. Area under y = √(4 − x²) from x = 0 to 2
Ο
68. Volume of solid when y = x from x = 0 to 2 revolved about x-axis
8Ο/3
69. Area between y = cos x and x-axis from x = 0 to Ο/2
1
70. Area under y = 1/x from x = 1 to e
1
71. Area under y = x³ from x = 1 to 2
15/4
72. Volume when y = x² from x = 0 to 1 revolved about x-axis
Ο/5
73. Area under y = x⁴ from x = 0 to 2
32/5
74. Area between y = x² and y = x from x = 0 to 1
1/6
75. Volume of solid when y = sin x from x = 0 to Ο revolved about x-axis
Ο²/2
76. Area under y = tan x from x = 0 to Ο/4
ln(√2)
77. Volume when y = √x, x = 0 to 1, revolved about x-axis
Ο/2
78. Area under y = e^x from x = 0 to 2
e² − 1
79. Area under y = cos x from x = 0 to Ο (signed)
0 (signed); absolute area = 2
80. Area under y = 1/x² from x = 1 to 3
2/3
81. Volume when y = x³ from x = 0 to 2 revolved about x-axis
128Ο/7
82. Area between y = x² and y = 3x from x = 0 to 3
9/2
83. Area under y = √(1 − x²) from x = 0 to 1
Ο/4
84. Volume when y = x² + 1, x = 0 to 1 revolved about x-axis
28Ο/15
85. Area under y = x² from x = −1 to 1
2/3
86. Area under y = 1/(1 + x) from x = 1 to 4
ln(5/2)
87. Volume when y = sin x, x = 0 to Ο/2, revolved about x-axis
Ο²/4
88. Area under y = x⁴ from x = 0 to 3
243/5
89. Area between y = x and y = x² from x = 0 to 1
1/6
90. Volume when y = √x from x = 0 to 9 revolved about x-axis
81Ο/2
91. Area under y = e^x from x = 0 to 1
e − 1
92. Area under y = ln x from x = 1 to 4
4 ln 4 − 3
93. Area under y = cos x from x = 0 to Ο/2
1
94. Area under y = 1/x from x = 1 to 2
ln 2
95. Volume when y = x³ from x = 0 to 1 revolved about x-axis
Ο/7
96. Area under y = x² + 1 from x = 0 to 2
14/3
97. Area under y = sin x from x = 0 to Ο
2
98. Area between y = x² and y = 4x − 4
0 (the curves are tangent at x = 2; no finite enclosed region)
99. Volume when y = √x from x = 0 to 4 revolved about x-axis
8Ο
100. Area under y = 1/x from x = 1 to e²
2
