Karnataka 2nd PUC Mathematics Application of Integrals MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: APPLICATIONS OF INTEGRALS

AREA UNDER SIMPLE CURVES
Area of the region bounded by the curve 𝒚 = 𝒇(𝒙), the X – axis and the lines x = a and x = b .
Case I: Area of the region when the curve is above the X – axis.

𝒃 𝒃
∴ Area (PQRSP) = ∫𝒂 𝒚𝒅𝒙 = ∫𝒂 𝒇(𝒙)𝒅𝒙

Case II: Area of the region when the curve is below the X – axis.

𝒃 𝒃
∴ Area = |∫𝒂 𝒚𝒅𝒙| = |∫𝒂 𝒇(𝒙)𝒅𝒙|

Case III: Area of the region when the curve is above and below the X – axis.

𝑪 𝒃
∴ Area = |𝑨𝟏 | + 𝑨𝟐 = |∫𝒂 𝒇(𝒙)𝒅𝒙| + ∫𝑪 𝒇(𝒙)𝒅𝒙

2026-27 MATHEMATICS CET MATERIAL Page 1 of 7

Area of the region bounded by the curve 𝒙 = 𝒈(𝒚), the Y – axis and the lines y = a and y = b .

𝒅 𝒅
∴ Area = ∫𝒄 𝒙𝒅𝒙 = ∫𝒄 𝒈(𝒚)𝒅𝒚

Some Standard Results

 The area of the circle 𝒙𝟐 + 𝒚𝟐 = 𝒂𝟐 is 𝝅𝒓𝟐 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝒙𝟐 𝒚𝟐
 The area of the ellipse 𝒂𝟐 + 𝒃𝟐 = 𝟏 is 𝝅𝒂𝒃 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.

 The area of the region bounded by the circle 𝒙𝟐 + 𝒚𝟐 = 𝒂𝟐 and the line
𝒂𝟐 (𝝅−𝟐)
𝒙 + 𝒚 = 𝒂 is given by 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟒
𝒙𝟐 𝒚𝟐
 The area of the region bounded by the ellipse 𝒂𝟐 + 𝒃𝟐 = 𝟏 and the line
𝒙 𝒚 𝒂𝒃(𝝅−𝟐)
+ 𝒃 = 𝟏 is given by 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝒂 𝟒

 The area of the region bounded by the parabola 𝒚𝟐 = 𝟒𝒂𝒙 and the line
𝟖𝒂𝟐
𝒚 = 𝒎𝒙 is given by 𝟑𝒎𝟑 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.

 The area of the region bounded by the parabola 𝒙𝟐 = 𝟒𝒂𝒚 and the line
𝟖𝒂𝟐 𝒎𝟑
𝒚 = 𝒎𝒙 is given by 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟑

 The area of the region bounded by the parabola 𝒚𝟐 = 𝟒𝒂𝒙 and the line
𝟖𝒎√𝒂𝒎
𝒙 = 𝒎 is given by 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟑

 The area of the region bounded by the parabola 𝒚𝟐 = 𝟒𝒂𝒙 (or 𝒙𝟐 = 𝟒𝒂𝒚 )
𝟖𝒂𝟐
and its latus rectum is 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟑

 The area of the region bounded by the parabolas 𝒚𝟐 = 𝟒𝒂𝒙 and
𝟏𝟔𝒂𝒃
𝒙𝟐 = 𝟒𝒃𝒚 is given by 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟑

 The area of the region bounded by one arc of 𝑦 = 𝒔𝒊𝒏(𝒂𝒙) or
𝟐
𝒚 = 𝒄𝒐𝒔(𝒂𝒙) and the 𝒙 − 𝒂𝒙𝒊𝒔 is given by |𝒂| 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.

 The area of the region bounded by the curve 𝑦 = 𝒔𝒊𝒏(𝒂𝒙) or
𝟐𝒏
𝒚 = 𝒄𝒐𝒔(𝒂𝒙) and the 𝒙 − 𝒂𝒙𝒊𝒔 in [0, 𝑛𝜋] is given by |𝒂| 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.

2026-27 MATHEMATICS CET MATERIAL Page 2 of 7

𝒙 𝟐/𝟑 𝒚 𝟐/𝟑 𝟑𝝅𝒂𝒃
 The area enclosed by the curve (𝒂) + (𝒃) = 𝟏 is 𝟖
𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟑𝝅𝒂𝟐
 The area of the asteroid 𝒙𝟐/𝟑 + 𝒚𝟐/𝟑 = 𝒂𝟐/𝟑 is 𝟖
𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.

 The area bounded by √𝒙 + √𝒚 = √𝒂 and the co – ordinate axes
𝒂𝟐
is 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠.
𝟔

 The area of the region bounded by 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄, (𝒃𝟐 − 𝟒𝒂𝒄 > 𝟎) and the 𝒙 − 𝒂𝒙𝒊𝒔
𝟑
(𝒃𝟐 −𝟒𝒂𝒄)𝟐
is 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠
𝟔𝒂𝟐

AREA UNDER SIMPLE CURVES
1. The area of the region bounded by the circle 𝒙𝟐 + 𝒚𝟐 = 𝟏 is (Easy)
(A) 𝜋 sq. units (B) 2𝜋 sq. units (C) 3𝜋 sq. units (D) 4𝜋 sq. units
2. The area of the circle 𝟐𝒙𝟐 + 𝟐𝒚𝟐 = 𝟏 in sq. units (Easy)
𝜋
(A) 𝜋 sq. units (B) √𝜋 sq. units (C) 2 sq. units (D) 2𝜋 sq. units

3. The area of the region bounded by the curve 𝒚 = √𝟏𝟔 − 𝒙𝟐 and x-axis is
[KCET 2021] (Average)
(A) 8𝜋 sq. units (B) 256𝜋 sq. units (C) 16𝜋 sq. units (D) 4𝜋 sq. units
4. Area lying in the first quadrant by the circle 𝒙𝟐 + 𝒚𝟐 = 𝟒 and the lines
𝒙 = 𝟎 𝒂𝒏𝒅 𝒙 = 𝟐 𝒊𝒔 (Average)
𝜋 𝜋 𝜋
(A) 4 sq. units (B) 3 sq. units (C) 2 sq. units (D) 𝜋 sq. units

5. Area of the region in the first quadrant enclosed by the x-axis, the line
𝒚 = 𝒙 and the circle 𝒙𝟐 + 𝒚𝟐 = 𝟑𝟐is (Average)
(A) 24𝜋 sq. units (B) 32𝜋 sq. units (C) 16𝜋 sq. units (D) 4𝜋 sq. units
6. Smaller area enclosed by the circle 𝒙𝟐 + 𝒚𝟐 = 𝟒 and the line 𝒙 + 𝒚 = 𝟐 is (Difficult)
(A) 2(𝜋 − 2) sq.units (B) (𝜋 − 2) sq.units
(C) (2𝜋 − 1) sq.units (D) 2(𝜋 + 2) sq.units
7. The area enclosed by the curve 𝒙 = √𝟑𝒄𝒐𝒔𝜽, 𝒚 = √𝟑𝒔𝒊𝒏𝜽 is [KCET 2026] (Difficult)
(A) √3𝜋 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠 (B) 9𝜋 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠 (C) 6𝜋 sq. units (D) 3𝜋 sq. units
𝒙𝟐 𝒚𝟐
8. The area of the region bounded by the ellipse 𝟐𝟓 + 𝟏𝟔 = 𝟏 is (Easy)

(A) 20𝜋 2 sq. units (B) 20𝜋 sq. units (C) 16𝜋 sq. units (D) 25𝜋 sq. units

2026-27 MATHEMATICS CET MATERIAL Page 3 of 7

𝒙𝟐 𝒚𝟐
9. If the area of the region bounded by the ellipse 𝟐𝟓 + 𝝀𝟐 = 𝟏 is 𝟐𝟎𝝅 square

units, then 𝝀 is [KCET 2021] (Average)
(A) ±4 (B) ±3 (C) ±2 (D) ±1
10. The area of the region bounded by the curve 𝟗𝒙𝟐 + 𝟒𝒚𝟐 = 𝟑𝟔 is (Easy)
(A) 4𝜋 sq. units (B) 6𝜋 sq. units (C) 9𝜋 sq. units (D) 36𝜋 sq. units
𝒙𝟐 𝒚𝟐
11. 𝑨𝑶𝑩 is a positive quadrant of the ellipse 𝒂𝟐 + 𝒃𝟐 = 𝟏, where 𝑶𝑨 = 𝒂,

𝑶𝑩 = 𝒃. The area between the arc 𝑨𝑩 and the chord 𝑨𝑩 of the ellipse is (Difficult)
(A) 𝜋𝑎𝑏 sq. units (B) (𝜋 − 2)𝑎𝑏 sq. units
𝑎𝑏(𝜋−2) 𝑎𝑏(𝜋+2)
(C) sq. units (D) sq. units
4 4

12. The area of the region bounded by the lines 𝒚 = 𝒎𝒙 , 𝒙 = 𝟏, 𝒙 = 𝟐, and
𝒙 axis is 6 sq. units, then 𝒎 is [KCET 2014] (Easy)
(A) 1 (B) 2 (C) 3 (D) 4
13. The area bounded by the line y = x, x-axis and ordinates x = – 1 and x = 2 is
[KCET 2018] (Average)
3 5
(A) 2 (B) 2 (C) 2 (D) 3

14. The area of the region bounded by the line 𝒚 = 𝟐𝒙 + 𝟏, x axis and
ordinates 𝒙 = −𝟏 & 𝒙 = 𝟏 is [KCET 2020] (Average)
5 9
(A) 2 sq. units (B) 2 sq. units (C) 5 sq. units (D) 4 sq. units

15. The area of the region bounded by the curve 𝒚 = 𝒙 + 𝟏 and the lines 𝒙 = 𝟑
and 𝒙 = 𝟓 is [KCET 2023] (Average)
7 11
(A) 2 sq. units (B) 2 sq. units (C) 7 sq. units (D) 10 sq. units

16. The area of the region bounded by the curve 𝒙 = 𝟐𝒚 + 𝟑 and the lines
𝒚 = 𝟏 and 𝒚 = −𝟏 is (Average)
3
(A) 4 sq. units (B) 2 sq. units (C) 6 sq. units (D) 8 sq. units

17. The area of the region bounded by 𝒚 = |𝒙|, 𝒙 ≤ 𝟓 in the first quadrant is (Average)
11 17 25 27
(A) 2 sq. units (B) 2 sq. units (C) 2 sq. units (D) 2 sq. units

18. The area bounded by the curve 𝒚 = 𝒙|𝒙|, 𝒙-axis and the ordinates
𝒙 = −𝟑 and 𝒙 = 𝟑 is (Difficult)
(A) 15 sq. units (B) 18 sq. units (C) 19 sq. units (D) 20 sq. units
19. Area bounded by 𝒚 = 𝒙𝟑 , 𝒚 = 𝟖 and 𝒙 = 𝟎 is …….. [KCET 2015] (Average)
(A) 12 sq. units (B) 2 sq. units (C) 6 sq. units (D) 14 sq. units

2026-27 MATHEMATICS CET MATERIAL Page 4 of 7

20. The area enclosed between the graph of 𝒚 = 𝒙𝟑 and the lines
𝒙 = 𝟎, 𝒚 = 𝟏, 𝒚 = 𝟖 is (Average)
45 45
(A) 14 sq. units (B) sq. units (C) 7 sq. units (D) sq. units
4 8

21. Area lying between the curves 𝒚𝟐 = 𝟐𝒙 and 𝒚 = 𝒙 is [KCET 2016](Average)
1 2 3 1
(A) 4 sq. units (B) 3 sq. units (C) 4 sq. units (D) 3 sq. units

22. The area of the region bounded by the curve 𝒚 = 𝒙𝟐 and the line y =16 is
[KCET 2017, 2025] (Average)
64 32 256 128
(A) 3 sq. units (B) 3 sq. units (C) sq. units (D) sq. units
3 3

23. The area of the region bounded by 𝒚𝟐 = 𝟖𝒙 and the straight line 𝒚 = 𝟐𝒙 is
(in sq. units). [KCET 2020] (Average)
8 16 4 3
(A) 3 (B) 3 (C) 3 (D) 4

24. Area lying between the curves 𝒚𝟐 = 𝟒𝒙 and 𝒚 = 𝟐𝒙 is (Average)
1 2 3 1
(A) 4 sq. units (B) 3 sq. units (C) 4 sq. units (D) 3 sq. units

25. The area of the region bounded by the parabola 𝒚𝟐 = 𝒙 and the straight
line 𝟐𝒚 = 𝒙 is (Average)
4 2 1
(A) 3 sq. units (B) 3 sq. units (C) 3 sq. units (D) 1 sq. units

26. The area of the region bounded by the line 𝒚 = 𝟑𝒙 and the curve 𝒚 = 𝒙𝟐 in
sq. units is [KCET 2024] (Average)
9
(A) 10 sq. units (B) 2 sq. units (C) 9 sq. units (D) 5 sq. units

27. The area of the region bounded by the curve 𝒚𝟐 = 𝒙𝟑 , the 𝒚 − 𝒂𝒙𝒊𝒔 and the
lines 𝒚 = 𝟏 and 𝒚 = 𝟖 is [KCET 2026] (Average)
93 155
(A) 93 sq. units (B) 5 sq. units (C) 155 sq. units (D) sq. units
3

𝒙𝟐 , 𝒙 < 𝟎
28. The area bounded by the curve 𝒚 = { and the (Difficult)
𝒙, 𝒙 ≥ 𝟎
line 𝒚 = 𝟒 is
8 32 16 40
(A) 3 sq. units (B) 3 sq. units (C) 3 sq. units (D) 3 sq. units

29. The area enclosed between the parabola 𝒚 = 𝒙𝟐 − 𝒙 + 𝟐 and the line
𝒚 = 𝒙 + 𝟐 in sq. units equals to (Average)
8 2 1 4
(A) 3 sq. units (B) 3 sq. units (C) 3 sq. units (D) 3 sq. units

2026-27 MATHEMATICS CET MATERIAL Page 5 of 7

30. The area bounded by the curve 𝒚 = 𝐥𝐨𝐠 𝒙, the 𝒙 − 𝒂𝒙𝒊𝒔 and the line 𝒙 = 𝒆
is given by (Difficult)
1 1
(A) 1 + (B) 𝑒 (C) 1 (D) 1 −
𝑒 𝑒

31. The area of the region bounded by the curve 𝒚 = 𝐬𝐢𝐧 𝒙 between the
𝝅
abscissa 𝒙 = 𝟎, 𝒙 = 𝟐 and the x-axis is (Average)

(A) 2 sq. units (B) 4 sq. units (C) 3 sq. units (D) 1 sq. unit
𝒙
32. The area bounded by the curve 𝒚 = 𝐬𝐢𝐧 (𝟑) , 𝒙 𝒂𝒙𝒊𝒔, the lines 𝒙 = 𝟎 and

𝒙 = 𝟑𝝅 is [KCET 2025] (Average)
(A) 9 sq. units (B) 0 sq. units (C) 6 sq. units (D) 3 sq. units
33. Area of the region bounded by the curve 𝒚 = 𝐜𝐨𝐬 𝒙 , 𝒙 = 𝟎 and 𝒙 = 𝝅 is
[KCET 2017, 18] (Average)
(A) 1 sq. units (B) 2 sq. units (C) 3 sq. units (D) 4 sq. units
34. Area of the region bounded by the curve 𝒚 = 𝒕𝒂𝒏𝒙, the 𝒙 axis and the line
𝝅
𝒙 = 𝟑 is [KCET 2022] (Average)
1
(A) log 2 (B) log2 (C) – log2 (D) 0
𝝅
35. The area bounded by the curve 𝒚 = 𝒔𝒆𝒄𝟐 𝒙, 𝒚 = 𝟎 and |𝒙| = 𝟑 is (Difficult)

(A) √3 sq. units (B) √2 sq. units (C) 2√3 sq. units (D) 3√2 sq. units
36. The area bounded by the curve 𝒚 = 𝟐𝒄𝒐𝒔𝒙 – 𝒔𝒊𝒏 𝒙, the 𝒙-axis and the
𝝅
ordinates 𝒙 = 𝟎, 𝒙 = 𝟒 in sq. units is (Average)
3 3 3
(A) − 1 (B) (C) 1 (D) 1 +
√2 √2 √2

37. The area of the region bounded by Y-axis, 𝒚 = 𝐜𝐨𝐬 𝒙 and 𝒚 = 𝐬𝐢𝐧 𝒙,
𝝅
𝟎 ≤ 𝒙 ≤ 𝟐 is [KCET 2019] (Difficult)

(A) (√2 + 1) sq. units (B) (√2 − 1) sq. units

(C) (2 − √2) sq. units (D) √2 sq. units
𝝅
38. In the interval (𝟎, 𝟐 ), area lying between the curves 𝒚 = 𝒕𝒂𝒏𝒙 and 𝒚 = 𝒄𝒐𝒕𝒙

and the X-axis is [KCET 2023] (Difficult)
(A) log2 sq. units (B) 2log2 sq. units
(C) 3log2 sq. units (D) 4 log2 sq. units

2026-27 MATHEMATICS CET MATERIAL Page 6 of 7

39. The value of the parameter ‘𝒂’ such that the area bounded by
𝒚 = 𝒂𝟐 𝒙𝟐 + 𝒂𝒙 + 𝟏, coordinate axes and the line 𝒙 = 𝟏 attains its least
value, is equal to (Difficult)
1 1 3
(A) − 4 (B) − 2 (C) −1 (D) − 4

40. The area bounded by 𝒇(𝒙) = 𝒔𝒊𝒏𝟐 𝒙 and the 𝒙- axis from 𝒙 = 𝒂 to 𝒙 = 𝒃,
where 𝒇′′ (𝒂) = 𝒇′′ (𝒃) = 𝟎 [∀ 𝒂, 𝒃 ∈ (𝟎, 𝝅)], is (Difficult)
𝜋 𝜋+2
(A) 2 (B) 𝜋 + 2 (C) 4 (D) 4

APPLICATIONS OF INTEGRALS
1 2 3 4 5 6 7 8 9 10
A C A D D B D B A B
11 12 13 14 15 16 17 18 19 20
C D C B D C C B A B
21 22 23 24 25 26 27 28 29 30
B C C D A B B D D C
31 32 33 34 35 36 37 38 39 40
D C B B C A B A D D

2026-27 MATHEMATICS CET MATERIAL Page 7 of 7