Applications of Integrals

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1847
The area bounded by the curve $y = x|x|$, x-axis and the ordinates $x = -1$ and $x = 1$ is given by
(A)
(B) $1/3$
(C) $2/3$
(D) $4/3$
MEDIUM
2 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1713
An ant is observed crawling on a sheet of paper along a straight line given by equation $y=2x-4$. Area of the surface covered by the ant bounded by y-axis, x-axis and $x=1$ is :
(A) 1 sq. unit
(B) 3 sq. units
(C) 2 sq. units
(D) 4 sq. units
MEDIUM
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1712
The area of the region bounded by the curve $y=x$ and x-axis, between $x=0$ and $x=2$ is:
(A) 2 sq. units
(B) $\frac{1}{2}$ sq. unit
(C) 1 sq. unit
(D) 4 sq. units
EASY
4 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1711
The area of the shaded region of the circle given below is equal to :
(A) $\int_{1}^{3}\sqrt{9-y^{2}}dy$
(B) $2\int_{1}^{3}\sqrt{9-y^{2}}dy$
(C) $\int_{0}^{3}\sqrt{9-x^{2}}dx$
(D) $2\int_{0}^{3}\sqrt{9-x^{2}}dx$
MEDIUM
5 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1710
Which of the following expressions will give the area of region bounded by the curve $y=x^{2}$ and line $y=16$?
(A) $\int_{0}^{4}x^{2}dx$
(B) $2\int_{0}^{4}x^{2}dx$
(C) $\int_{0}^{16}\sqrt{y}dy$
(D) $2\int_{0}^{16}\sqrt{y}dy$
MEDIUM
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#642
The area of the shaded region (figure) represented by the curves \(y=x^{2}\), \(0\le x\le2\) and y-axis is given by
(A) \(\int_{0}^{2} x^2 dx\)
(B) \(\int_{0}^{2} \sqrt{y} dy\)
(C) \(\int_{0}^{4} x^2 dx\)
(D) \(\int_{0}^{4} \sqrt{y} dy\)
MEDIUM
7 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#641
The area of the region enclosed by the curve \(y=\sqrt{x}\) and the lines \(x=0\) and \(x=4\) and x-axis is:
(A) \(\frac{16}{9}\) sq. units
(B) \(\frac{32}{9}\) sq. units
(C) \(\frac{16}{3}\) sq. units
(D) \(\frac{32}{3}\) sq. units
MEDIUM
8 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#640
The area of the region enclosed between the curve \(y=x|x|\), x-axis, \(x=-2\) and \(x=2\) is:
(A) \(\frac{8}{3}\)
(B) \(\frac{16}{3}\)
(C) 0
(D) 8
MEDIUM
9 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#638
The area of the shaded region bounded by the curve \(y^2=x,x=4\) and the \(x-\)axis is given by

(A) \( \int_{0}^{4} x \, dx\)
(B) \( \int_{0}^{2} y^2 \, dy\)
(C) \( 2 \int_{0}^{4} \sqrt{x} \, dx\)
(D) \( \int_{0}^{4} \sqrt{x} \, dx\)
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#637
The area of the region bounded by the curve \(y^{2}=x\) between \(x=0\) and \(x=1\) is:
(A) \(\frac{3}{2}\) sq units
(B) \(\frac{2}{3}\) sq units
(C) 3 sq units
(D) \(\frac{4}{3}\) sq units
MEDIUM
11 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#639
Area of the region bounded by curve \(y^{2}=4x\) and the X-axis between \(x=0\) and \(x=1\) is:
(A) \(\frac{2}{3}\)
(B) \(\frac{8}{3}\)
(C) 3
(D) \(\frac{4}{3}\)
MEDIUM

Very Short Answer (2 Mark)

1 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1404
Calculate the area of the region bounded by the curve $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the x-axis using integration.
MEDIUM
2 Competency APPLY VSA 2 Mark(s) 2023
#872
Sketch the region bounded by the lines 2x+y=8, y=2, y=4 and the y-axis. Hence, obtain its area using integration.
MEDIUM

Short Answer (3 Marks)

1 Competency UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1370
Sketch the graph of $y=|x+3|$ and find the area of the region enclosed by the curve, x-axis, between $x=-6$ and $x=0$, using integration.
MEDIUM
2 Competency APPLY SA 3 Mark(s) 2023
#873
Find the area of the following region using integration: {(x,y): y² ≤ 2x and y ≥ x-4}
HARD

Long Answer (5 Marks)

1 Knowledge APPLY LA 5 Mark(s) 2026-27 CBSE Sample QP
#2098
Using integration, find the area of the region bounded by the curve $y^2 = 2x$ and the line $x - 2y = 0$.
2 Competency APPLY LA 5 Mark(s) 2026-27 CBSE Sample QP
#2097
Using Integration, find the area of the region $\{(x, y) : 0 \leq y \leq \sqrt{4 - x^2}, x \geq 2 - y\}$.
HARD
3 Competency APPLY LA 5 Mark(s) 2026 AISSCE(Board Exam)
#1820
Sketch the curve $\{(x, y): 100x^{2}+25y^{2}=2500\}$ and find the area of the region enclosed by it, using integration.
MEDIUM
4 Competency APPLY LA 5 Mark(s) 2026 AISSCE(Board Exam)
#1819
Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between $x=4$ and $x=8$.
MEDIUM
5 Competency UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1483
A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and discovered the scratch passing through the origin inclined at an angle $\frac{\pi}{4}$ anticlockwise along the positive direction of x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration.
MEDIUM
6 Competency REMEMBER LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1461
Draw a rough sketch for the curve $y=2+|x+1|$. Using integration, find the area of the region bounded by the curve $y=2+|x+1|$, $x=-4$, $x=3$ and $y=0$.
MEDIUM
7 Competency UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1437
Sketch a graph of $y=x^{2}$. Using integration, find the area of the region bounded by $y=9$, $x=0$ and $y=x^{2}$.
MEDIUM
8 Knowledge REMEMBER LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1393
Using integration, find the area of the region bounded by the line $y=5x+2$, the x-axis and the ordinates $x=-2$ and $x=2$.
MEDIUM
9 Knowledge UNDERSTAND LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1351
Find the area of the region bounded by the curve $4x^{2}+y^{2}=36$ using integration.
MEDIUM
10 Knowledge UNDERSTAND LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1327
Using integration, find the area bounded by the ellipse $9x^{2}+25y^{2}=225$, the lines $x=-2,$ $x=2$, and the X-axis.
MEDIUM
11 Competency REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1326
Sketch the graph of $y=x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x=-2$ and $x=2,$ using integration.
MEDIUM
12 Knowledge REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1308
Using integration, find the area of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1,$ included between the lines $x=-2$ and $x=2$.
MEDIUM
13 Knowledge REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1287
If $A_{1}$ denotes the area of region bounded by $y^{2}=4x,$ $x=1$ and x-axis in the first quadrant and $A_{2}$ denotes the area of region bounded by $y^{2}=4x,$ $x=4$, find $A_{1}:A_{2}$.
MEDIUM
14 Knowledge REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1263
Using integration, find the area of the region enclosed between the circle $x^{2}+y^{2}=16$ and the lines $x=-2$ and $x=2.$
MEDIUM
15 Competency APPLY LA 5 Mark(s) 2023
#930
The area of the region bounded by the line \(y=mx (m>0)\), the curve \(x^{2}+y^{2}=4\) and the \(x\)-axis in the first quadrant is \(\frac{\pi}{2}\) units. Using integration, find the value of m.
HARD
16 Competency APPLY LA 5 Mark(s) 2023
#929
33. Using integration, find the area of the region bounded by the parabola $y^{2}=4ax$ and its latus rectum.
MEDIUM
17 Competency APPLY LA 5 Mark(s) 2023
#874
Find the area of the region bounded by the curves x^{2}=y, y=x+2 and x-axis, using integration.
HARD

Subjective / Numerical

1 Competency APPLY SUBJECTIVE 4 Mark(s) 2026 AISSCE(Board Exam)
#2100
Roundabouts are often made on busy roads to ease the traffic and avoid red lights.One such round-about is made such that equation representing its boundary is given by $C_1 : x^2 + y^2 = 64$. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_2 : x^2 + y^2 = 4$.
Based on the given information, answer the following questions :
(i) Represent the given equations $C_1$ and $C_2$ with the help of a diagram. [1 mark]
(ii) Express y as a function of $x$, ($y = f(x)$), for both $C_1$ an $C_2$. [1 mark]
(iii) (a) Using integration find the area of region covered by the roundabout. [2 marks]
OR
(iii) (b) Using integration, find the area of region covered by circular pond. [2 marks]
MEDIUM