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CBSE · Class 12 · Mathematics

Application of Integrals

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The area bounded by y = x2, the x-axis and the line x = 3 is:
  1. 9/2 square units
  2. 27 square units
  3. 27/2 square units
  4. 9 square units
2.
The area enclosed by the ellipse x2/81 + y2/64 = 1 is:
  1. 5184π square units
  2. 145π square units
  3. 72π square units
  4. 18π square units
3.
The area of the region bounded by the parabola y2 = 4x and the line x = 1 is:
  1. 8 square units
  2. 16/3 square units
  3. 4/3 square units
  4. 8/3 square units
4.
The area enclosed by the parabola y = x2 and the line y = 36 is:
  1. 288 square units
  2. 144 square units
  3. 1296 square units
  4. 576 square units
5.
The area bounded by y = f(x) ≥ 0, the x-axis and x = a, x = b is:
  1. ∫ from a to b of f(x) dx
  2. ∫ from a to b of x dx
  3. f′(b) − f′(a)
  4. f(b) − f(a)
6.
The area of the region in the first quadrant enclosed by the circle x2 + y2 = 36 and the coordinate axes is:
  1. 36π square units
  2. 9π square units
  3. 18π square units
  4. 9 square units
7.
The area bounded by y = x3, the x-axis and the lines x = 0 and x = 5 is:
  1. 625/4 square units
  2. 625/3 square units
  3. 125/3 square units
  4. 625 square units
8.
Assertion (A): The area enclosed by the ellipse x2/16 + y2/49 = 1 is 28π square units.
Reason (R): The area enclosed by an ellipse with semi-axes a and b is π(a2 + b2).
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Find the area bounded by the curve y = x2 + 6x + 15, the x-axis and the lines x = −5 and x = −1.
10.
Find the area of the region bounded by the parabola y2 = 64x and the line x = 16.
11.
Find the area enclosed by the ellipse x2/81 + y2/64 = 1.
12.
Find the area of the region bounded by the line y = x, the x-axis and the line x = 7.
13.
Find the area bounded by y = |x − 5|, the x-axis and the lines x = 0 and x = 6.
14.
Find the area of the region above the x-axis enclosed by the circle x2 + y2 = 64.
15.
Find the area bounded by y = √(9x), the x-axis and the line x = 1.
16.
Find the area bounded by y = x3, the x-axis and the lines x = 2 and x = 4.
Section C — Word problems4 × 4 = 16 marks
17.
Using integration, find the area of the triangle with vertices (−2, 0), (4, 1), (5, 6).
18.
Find the area of the region enclosed by the parabola y = x2 and the line y = x + 6.
19.
Find the area of the smaller region bounded by the ellipse x2/81 + y2/9 = 1 and the line x/9 + y/3 = 1.
20.
Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 4.
AmbifulWorksheets
Practice that never repeats
CBSE · Class 12 · Mathematics

Answer key

Application of Integrals · AW-12M-AOI-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) 9 square units∫03 x2 dx = 1 × 27/3
2(c) 72π square unitsπab with a = 9, b = 8
3(d) 8/3 square units2∫01 2√x dx = (8/3) × 1 × 1
4(a) 288 square units2∫06 (36 − x2) dx
5(a) ∫ from a to b of f(x) dx—
6(b) 9π square unitsa quarter of π × 6 × 6
7(a) 625/4 square units[x4/4] from 0 to 5
8(c) A is true, but R is false.A is true: πab = π × 4 × 7 = 28π. R is false.
988/3 square unitsthe curve stays above the x-axis; ∫−5−1 (x2 + 6x + 15) dx
102048/3 square units2∫016 8√x dx = (8/3) × 4 × 64
1172π square units4 × (8/9)∫09 √(81 − x2) dx = 4 × (8/9) × 81π/4
1249/2 square units∫07 x dx
1313 square unitstwo triangles: 25/2 + 1/2
1432π square units2∫08 √(64 − x2) dx = half of π × 64
152 square units∫01 √(9x) dx = (2/3) × √9 × 1√1 = (2/3) × 3
1660 square units[x4/4] from 2 to 4
1729/2 square unitswrite the equation of each side, integrate between the vertices' x-values, and add or subtract the areas
18125/6 square unitsthey meet at x = −2 and x = 3; ∫−23 (x + 6 − x2) dx
19(27/4)(π − 2) square unitsquarter ellipse 27π/4 minus the triangle 27/2
20π/2 square unitsthe line meets the circle at x = 2/√2; the region is an eighth of the circle