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Sunday, 25 October

JEE Main · Maths · Class 12

Application of Integrals: JEE Main previous year questions

91 questions in 86 of the 108 JEE Main shifts Lumi analysed, asked in 8 of 8 years. Here is what this chapter tests and how.

Study this chapter
Questions
91
in 86 of 108 shifts · 8 of 8 years
Average a shift
0.8
of 25 Maths · 0.9 since 2024 0.1
Typical range
0.6–0.9
questions a shift; fewest 0.5, most 1.0
Weighted estimate
0.8
model, recent years count more · about 3.4 marks
  • Typical range (fact)
  • Average a shift (fact)
  • Weighted estimate (model)

Questions are counted per shift, out of the 25 each subject has, in the shifts Lumi analysed; papers that offered 30 a subject (2017, 2021–2024) are scaled to 25. Typical range: the middle six of the eight years (2017, 2020–2026), leaving out the single highest and lowest year. The average and the range are historical fact. The weighted estimate is a model: each year's share, with recent years counting more (half-life three years), and chapters NTA dropped from the syllabus set near zero. It is a guide to effort, not a forecast of the paper.

What to do

How to spend time on Application of Integrals

Priority

Lower priority

One of the 7 Maths chapters that shared the last fifth of Maths questions in the 108 shifts analysed. How bands are set

Past questions
91in 86 of 108 shifts
Study plan gives it
16 hover 12 months; 12 h over 6
Study this chapter →

Ideas asked most often

  1. Area Between Curves22×
  2. Area of Inequalities15×
  3. Area Involving Max/Min Functions8×
  4. Area with Modulus7×
  5. Area Between Two Standard Parabolas6×

32 of the 65 ideas in Lumi’s map of this chapter have not been asked or touched in the shifts analysed. Of its idea-tagged questions from 2024 on, 9% tested an idea not asked in an earlier year, so learn the whole chapter, not just the list above.

Within each subject, chapters in NTA's 2026 syllabus are ranked by their average questions a shift over all 8 years analysed (history, not the model). Going down that list, the chapters that together held the first half of the subject's questions are Must do; the next ones, up to 80%, are Should do; the rest are Lower priority.

Key findings

Application of Integrals at a glance

  1. 01

    91%

    of its 34 questions from 2024 on that Lumi matched to an idea tested one JEE Main had asked in an earlier year; 100% came from a topic asked in an earlier year.

  2. 02

    49%

    of the 65 ideas in Lumi's map of the chapter have not been asked or touched on in the shifts analysed.

  3. 03

    37%

    of its questions asked for a numerical answer, 1.3 times the Maths average.

  4. 04

    32%

    of its questions were rated hard, against 26% across all Maths questions in the same shifts.

Questions a shift, year by year

Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.

0240.9’170.6’200.5’210.7’220.9’230.9’240.8’251’26

Raw count: 2017: 1 in 1 shift · 2020: 7 in 12 shifts · 2021: 15 in 24 shifts · 2022: 21 in 24 shifts · 2023: 13 in 12 shifts · 2024: 19 in 18 shifts · 2025: 6 in 8 shifts · 2026: 9 in 9 shifts.

Kind of question

SHARE OF QUESTIONS
  • Numerical answer37%
  • Multi-step30%
  • Calculation29%
  • Application4%

Most common asks: area of region defined by inequalities (18); area enclosed between two curves (8); area bounded by modulus curve and line (3).

Profile

This chapter in numbers

Where it ranks, how steady it is, and what it costs to study.

Rank, all JEE chapters
#49
of 70 by weighted estimate
Rank in Maths
#17
of 21 chapters
Share of Maths
3.1%
of its questions, all shifts analysed
Busiest year
2026
1.0 questions a shift
Year to year
Steady
varies by about 0.1 questions a shift
Long-run direction
+0.02
questions a shift, per year
Easy questions
12%
32% rated hard
Numerical answer
37%
1.3× the Maths rate
With a figure
0%
graph, diagram or structure
Study time
16 h
12 h on a 6-month plan
Marks per study hour
0.22
#59 of 70 chapters
Foundation for others
39%
of its ideas are used in other chapters

Difficulty, year by year

Each question rated easy, medium or hard. Each bar is that year’s split, so years compare directly; the number is the questions in that year’s analysed shifts (1 to 24 shifts a year).

  • 20171
  • 20207
  • 202115
  • 202221
  • 202313
  • 202419
  • 20256
  • 20269
  • Easy
  • Medium
  • Hard

What it asks you to do

SHARE OF QUESTIONS
  • Multi-step49%
  • Calculation46%
  • Application4%

Coverage

How much of the chapter JEE Main has used

We split the chapter into 65 ideas. In the 108 shifts analysed, JEE Main built a question around 24 of them. Most of the chapter has never been the main idea of a question, so past questions alone leave most of it out.

Asked as the main idea
24
Touched on in a question
9
Never asked
32

Topics

What repeats, topic by topic

This chapter's questions per topic in each year's analysed shifts. The number is the count; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. 1 questions had no close topic in Lumi's map and count for the chapter only.

’17’20’21’22’23’24’25’26
020+questions in that year’s analysed shifts, shaded per shift
  • Area Under Curves

    As a topic: 94 questions in 8 of 8 years, most often single correct, 36% numerical answer. Recent asks: area bounded by curve, line and axis; area of region from system of inequalities; numerical: area between curves from defined function.

  • Variable Separable Form

    As a topic: 42 questions in 7 of 8 years, most often single correct, 19% numerical answer.

Drill down

Topic, idea, question

Open a topic to see every idea JEE Main tested inside it, then open an idea to read the questions. Each row shows its 8 years.

  • Open89 · 98%71520121969

    22 ideas inside

    • OpenRepeated22 · 25%234255
    • OpenRepeated15 · 17%33222
    • OpenRepeated8 · 9%22
    • OpenRepeated7 · 8%32
    • OpenRepeated6 · 7%22
    • OpenRepeated6 · 7%22
    • OpenRepeated3 · 3%2
    • OpenRepeated3 · 3%
    • Open2 · 2%2
    • OpenRepeated2 · 2%
    • OpenRepeated2 · 2%
    • OpenRepeated2 · 2%
    • OpenRepeated2 · 2%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
    • Open1 · 1%
  • Open1 · 1%

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Ideas

The exact ideas JEE Main has asked

Inside the topics, 23 different ideas were tested; 13 came more than once. These are the repeats, then the ideas new since 2024.

Dots are the 8 years with analysed shifts, 2017 to 2026.

Every question

All 91 Application of Integrals questions

Newest first, shift by shift. The newest 40 show their opening words; older ones name the idea they tested. Each opens with its options and the official answer.

JEE Main 20269 questions

  1. 2 Apr 2026, Shift 2 · Q24If the area of the region bounded by 16x2−9y2=14416x^2 - 9y^2 = 144 and 8x−3y=248x - 3y = 24 is AA, then 3(A+6log⁡e(3))3(A + 6\log_e(3)) is equal toHardNumerical value
  2. 4 Apr 2026, Shift 1 · Q18The area of the region {(x,y):y≤π−∣x∣, y≤∣xsin⁡x∣, y≥0}\{(x,y): y\le\pi-|x|,\ y\le|x\sin x|,\ y\ge0\} is:HardSingle correct
  3. 4 Apr 2026, Shift 2 · Q18The area of the region bounded by the curves x+3y2=0x+3y^2=0 and x+4y2=1x+4y^2=1 is equal to:EasySingle correct
  4. 5 Apr 2026, Shift 1 · Q16The area of the region R={(x,y):xy≤27, 1≤y≤x2}R = \{(x, y) : xy \leq 27,\ 1 \leq y \leq x^2\} is equal to:MediumSingle correct
  5. 5 Apr 2026, Shift 2 · Q24Let f:R→Rf:\mathbf{R}\to\mathbf{R} be a function such that f(x)+3f(π2−x)=sin⁡xf(x)+3f\left(\frac{\pi}{2}-x\right)=\sin x, x∈Rx \in \mathbf{R}. Let the maximum…HardNumerical value
  6. 6 Apr 2026, Shift 1 · Q19The area of the region {(x,y):0≤y≤6−x, y2≥4x−3, x≥0}\{(x,y): 0\le y\le 6-x,\ y^2\ge 4x-3,\ x\ge 0\} is:MediumSingle correct
  7. 6 Apr 2026, Shift 1 · Q20Let ee be the base of natural logarithm and let f:{1,2,3,4}→{1,e,e2,e3}f:\{1,2,3,4\}\to\{1,e,e^2,e^3\} and…HardSingle correct
  8. 6 Apr 2026, Shift 2 · Q19The area of the region {(x,y):x2−8x≤y≤−x}\{(x,y): x^2-8x\le y\le -x\} is :EasySingle correct
  9. 8 Apr 2026, Shift 2 · Q18Let f:(1,∞)→Rf : (1, \infty) \to \mathbf{R} be a function defined as f(x)=x−1x+1f(x) = \frac{x-1}{x+1}. Let fi+1(x)=f(fi(x))f^{i+1}(x) = f\left(f^{i}(x)\right),…HardSingle correct

JEE Main 20256 questions

  1. 2 Apr 2025, Shift 1 · Q21If the area of the region {(x,y):∣4−x2∣≤y≤x2, y≤4, x≥0}\left\{(x, y): \left|4-x^2\right|\le y\le x^2,\ y\le4,\ x\ge0\right\} is…HardNumerical value
  2. 3 Apr 2025, Shift 1 · Q25The area of the region bounded by the curve y=max⁡{∣x∣,x∣x−2∣}y = \max\{|x|, x|x-2|\}, the xx-axis and the lines x=−2x = -2 and x=4x = 4 is equal to ______MediumNumerical value
  3. 3 Apr 2025, Shift 2 · Q19The area of the region {(x,y):∣x−y∣≤y≤4x}\left\{(x, y) : |x - y| \leq y \leq 4\sqrt{x}\right\} isMediumSingle correct
  4. 7 Apr 2025, Shift 1 · Q19If the area of the region bounded by the curves y=4−x24y = 4 - \frac{x^2}{4} and y=x−42y = \frac{x-4}{2} is equal to α\alpha, then 6α6\alpha equalsEasySingle correct
  5. 7 Apr 2025, Shift 2 · Q20If the area of the region {(x,y):1+x2≤y≤min⁡{x+7, 11−3x}}\{(x,y): 1+x^2\le y\le \min\{x+7,\ 11-3x\}\} is AA, then 3A3A is equal toMediumSingle correct
  6. 8 Apr 2025, Shift 2 · Q25Let the area of the bounded region {(x,y):0≤9x≤y2,y≥3x−6}\{(x, y) : 0 \le 9x \le y^2, y \ge 3x - 6\} be AA. Then 6A6A is equal to __________MediumNumerical value

JEE Main 202419 questions

  1. 4 Apr 2024, Shift 1 · Q13One of the points of intersection of the curves y=1+3x−2x2y=1+3x-2x^2 and y=1xy=\frac{1}{x} is (12,2)\left(\frac{1}{2}, 2\right). Let the area of the…HardSingle correct
  2. 4 Apr 2024, Shift 2 · Q11The area (in sq. units) of the region described by {(x,y):y2≤2x, and y≥4x−1}\{(x,y): y^2\le 2x,\ \text{and}\ y\ge 4x-1\} isMediumSingle correct
  3. 5 Apr 2024, Shift 1 · Q27The area of the region enclosed by the parabolas y=x2−5xy=x^2-5x and y=7x−x2y=7x-x^2 is ________.EasyNumerical value
  4. 5 Apr 2024, Shift 2 · Q12The area enclosed between the curves y=x∣x∣y = x|x| and y=x−∣x∣y = x - |x| is :MediumSingle correct
  5. 6 Apr 2024, Shift 1 · Q12Let the area of the region enclosed by the curves y=3xy=3x, 2y=27−3x2y=27-3x and y=3x−xxy=3x-x\sqrt{x} be AA. Then 10A10A is equal toMediumSingle correct
  6. 6 Apr 2024, Shift 2 · Q11If the area of the region {(x,y):ax2≤y≤1x, 1≤x≤2, 0<a<1}\left\{(x,y):\frac{a}{x^2}\le y\le\frac{1}{x},\ 1\le x\le 2,\ 0<a<1\right\} is (log⁡e2)−17(\log_e 2)-\frac{1}{7} then…MediumSingle correct
  7. 8 Apr 2024, Shift 1 · Q27Let the area of the region enclosed by the curve y=min⁡{sin⁡x,cos⁡x}y=\min\{\sin x,\cos x\} and the xx-axis between x=−πx=-\pi to x=πx=\pi be AA. Then A2A^2…MediumNumerical value
  8. 8 Apr 2024, Shift 2 · Q12The area of the region in the first quadrant inside the circle x2+y2=8x^2 + y^2 = 8 and outside the parabola y2=2xy^2 = 2x is equal to :MediumSingle correct
  9. 9 Apr 2024, Shift 1 · Q9The parabola y2=4xy^2=4x divides the area of the circle x2+y2=5x^2+y^2=5 in two parts. The area of the smaller part is equal to :MediumSingle correct
  10. 9 Apr 2024, Shift 2 · Q10The area (in square units) of the region enclosed by the ellipse x2+3y2=18x^2+3y^2=18 in the first quadrant below the line y=xy=x isMediumSingle correct
  11. 1 Feb 2024, Shift 1 · Q9The area enclosed by the curves xy+4y=16xy+4y=16 and x+y=6x+y=6 is equal to :MediumSingle correct
  12. 1 Feb 2024, Shift 2 · Q26The sum of squares of all possible values of kk, for which area of the region bounded by the parabolas 2y2=kx2y^2=kx and ky2=2(y−x)ky^2=2(y-x) is…HardNumerical value
  13. 1 Feb 2024, Shift 2 · Q28Three points O(0,0)O(0,0), P(a,a2)P(a,a^2), Q(−b,b2)Q(-b,b^2), a>0a>0, b>0b>0, are on the parabola y=x2y=x^2. Let S1S_1 be the area of the region bounded by…HardNumerical value
  14. 27 Jan 2024, Shift 1 · Q26Let the area of the region {(x,y):x−2y+4≥0, x+2y2≥0, x+4y2≤8, y≥0}\{(x,y): x-2y+4\geq0,\ x+2y^2\geq0,\ x+4y^2\leq8,\ y\geq0\} be mn\dfrac{m}{n}, where m and n are coprime…MediumNumerical value
  15. 29 Jan 2024, Shift 1 · Q25The area (in sq. units) of the part of the circle x2+y2=169x^2+y^2=169 which is below the line 5x−y=135x-y=13 is…HardNumerical value
  16. 30 Jan 2024, Shift 1 · Q11The area (in square units) of the region bounded by the parabola y2=4(x−2)y^2=4(x-2) and the line y=2x−8y=2x-8, is :MediumSingle correct
  17. 30 Jan 2024, Shift 2 · Q26The area of the region enclosed by the parabola (y−2)2=x−1(y-2)^2 = x - 1, the line x−2y+4=0x - 2y + 4 = 0 and the positive coordinate axes is ________.MediumNumerical value
  18. 31 Jan 2024, Shift 1 · Q10The area of the region {(x,y):y2≤4x, x<4, xy(x−1)(x−2)(x−3)(x−4)>0, x≠3}\left\{(x,y):y^2\leq4x,\ x<4,\ \frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,\ x\neq3\right\} isHardSingle correct
  19. 31 Jan 2024, Shift 2 · Q11The area of the region enclosed by the parabolas y=4x−x2y=4x-x^2 and 3y=(x−4)23y=(x-4)^2 is equal toMediumSingle correct

JEE Main 202313 questions

  1. 6 Apr 2023, Shift 1 · Q22Let the point (p,p+1)(p,p+1) lie inside the region E={(x,y):3−x≤y≤9−x2, 0≤x≤3}E=\left\{(x,y):3-x\le y\le\sqrt{9-x^2},\,0\le x\le3\right\}. If the set of all values of…MediumNumerical value
  2. 6 Apr 2023, Shift 1 · Q26If the area of the region S={(x,y):2y−y2≤x2≤2y, x≥y}S=\{(x,y):2y-y^2\le x^2\le 2y,\ x\ge y\} is equal to n+2n+1−πn−1\frac{n+2}{n+1}-\frac{\pi}{n-1}, then the natural…HardNumerical value
  3. 6 Apr 2023, Shift 2 · Q14The area bounded by the curves y=∣x−1∣+∣x−2∣y=|x-1|+|x-2| and y=3y=3 is equal toEasySingle correct
  4. 8 Apr 2023, Shift 1 · Q12The area of the region {(x,y):x2≤y≤8−x2, y≤7}\{(x,y):x^2\le y\le 8-x^2,\ y\le 7\} isMediumSingle correct
  5. 8 Apr 2023, Shift 2 · Q27Let the area enclosed by the lines x+y=2x + y = 2, y=0y = 0, x=0x = 0 and the curve f(x)=min⁡{x2+34,1+[x]}f(x) = \min\left\{x^2 + \frac{3}{4}, 1 + [x]\right\}…HardNumerical value
  6. 10 Apr 2023, Shift 1 · Q28Let y=p(x)y=p(x) be the parabola passing through the points (−1,0)(-1,0), (0,1)(0,1) and (1,0)(1,0). If the area of the region…HardNumerical value
  7. 10 Apr 2023, Shift 2 · Q26Area with Modulus · Medium · Numerical value
  8. 11 Apr 2023, Shift 1 · Q9Area of Inequalities · Medium · Single correct
  9. 11 Apr 2023, Shift 2 · Q25Area Involving Tangents and Normals · Medium · Numerical value
  10. 12 Apr 2023, Shift 1 · Q10Area Between Cubic Curve and Tangent · Medium · Single correct
  11. 13 Apr 2023, Shift 1 · Q12Area Involving Max/Min Functions · Medium · Single correct
  12. 13 Apr 2023, Shift 2 · Q11Area of Inequalities · Hard · Single correct
  13. 15 Apr 2023, Shift 1 · Q27Area Between Curves · Hard · Numerical value

JEE Main 202221 questions

  1. 25 Jul 2022, Shift 1 · Q8Area Involving Max/Min Functions · Medium · Single correct
  2. 26 Jul 2022, Shift 1 · Q5Area with Y-Axis · Medium · Single correct
  3. 26 Jul 2022, Shift 2 · Q20Area with Modulus · Medium · Single correct
  4. 27 Jul 2022, Shift 1 · Q9Area Between Curves · Hard · Single correct
  5. 27 Jul 2022, Shift 2 · Q8Area Between Curves · Medium · Single correct
  6. 28 Jul 2022, Shift 2 · Q70Exponential and Logarithmic Bounded Regions · Hard · Single correct
  7. 29 Jul 2022, Shift 1 · Q16Area with Modulus · Medium · Single correct
  8. 30 Jul 2022, Shift 2 · Q74Implicit Solution Curve Analysis · Hard · Single correct
  9. 30 Jul 2022, Shift 2 · Q85Area Between Curves · Hard · Numerical value
  10. 24 Jun 2022, Shift 1 · Q29Ratio of Divided Areas · Hard · Numerical value
  11. 24 Jun 2022, Shift 2 · Q26Area Between Parabola and Line · Easy · Numerical value
  12. 25 Jun 2022, Shift 2 · Q6Area Between Two Standard Parabolas · Medium · Single correct
  13. 26 Jun 2022, Shift 2 · Q7Area Between Two Standard Parabolas · Medium · Single correct
  14. 27 Jun 2022, Shift 1 · Q26Area of Inequalities · Hard · Numerical value
  15. 27 Jun 2022, Shift 2 · Q27Area of Inequalities · Hard · Numerical value
  16. 28 Jun 2022, Shift 1 · Q9Area of Inequalities · Medium · Single correct
  17. 28 Jun 2022, Shift 2 · Q9Area with Modulus · Medium · Single correct
  18. 29 Jun 2022, Shift 1 · Q6Area Between Curves · Medium · Single correct
  19. 29 Jun 2022, Shift 2 · Q8Optimal Strip Selection (dx vs dy) · Medium · Single correct
  20. 29 Jun 2022, Shift 2 · Q25Area of Ellipse and Sectors · Medium · Numerical value
  21. 30 Jun 2022, Shift 1 · Q24Finding Unknown Parameters from Area · Hard · Numerical value

JEE Main 202115 questions

  1. 3 Aug 2021, Shift 1 · Q87Area Between Parabola and Line · Easy · Numerical value
  2. 3 Aug 2021, Shift 2 · Q85Area Between Curves · Medium · Numerical value
  3. 26 Aug 2021, Shift 1 · Q85Area Between Parabola and Line · Medium · Numerical value
  4. 26 Aug 2021, Shift 2 · Q85Area with X-Axis · Medium · Numerical value
  5. 27 Aug 2021, Shift 2 · Q71Area Involving Tangents and Normals · Medium · Single correct
  6. 31 Aug 2021, Shift 2 · Q87Bisection of Bounded Areas · Medium · Numerical value
  7. 24 Feb 2021, Shift 1 · Q71Area Between Curves · Medium · Single correct
  8. 24 Feb 2021, Shift 2 · Q80Area Between Two Standard Parabolas · Easy · Single correct
  9. 25 Feb 2021, Shift 1 · Q88Area Between Trigonometric Curves · Medium · Numerical value
  10. 26 Feb 2021, Shift 2 · Q63Area Between Trigonometric Curves · Medium · Single correct
  11. 22 Jul 2021, Shift 2 · Q83Regions with Multiple Bounding Curves · Hard · Numerical value
  12. 25 Jul 2021, Shift 1 · Q63Area of Inequalities · Easy · Single correct
  13. 16 Mar 2021, Shift 2 · Q80Area Between Curves · Hard · Single correct
  14. 17 Mar 2021, Shift 2 · Q87Area Involving Max/Min Functions · Medium · Numerical value
  15. 18 Mar 2021, Shift 2 · Q70Area of Algebraic Curve Loops · Hard · Single correct

JEE Main 20207 questions

  1. 8 Jan 2020, Shift 1 · Q61Bisection of Bounded Areas · Hard · Single correct
  2. 8 Jan 2020, Shift 2 · Q62Area Between Parabola and Line · Easy · Single correct
  3. 9 Jan 2020, Shift 2 · Q63Area Between Curves · Medium · Single correct
  4. 2 Sep 2020, Shift 1 · Q62Area of Ellipse and Sectors · Medium · Single correct
  5. 2 Sep 2020, Shift 2 · Q62Bisection of Bounded Areas · Medium · Single correct
  6. 3 Sep 2020, Shift 1 · Q61Area of Inequalities · Medium · Single correct
  7. 6 Sep 2020, Shift 2 · Q62Area Between Curves · Easy · Single correct

JEE Main 20171 question

  1. 2 Apr 2017 (offline) · Q76Area of Inequalities · Hard · Single correct

Questions from the official JEE Main papers published by NTA. Chapter, topic and idea tags, counts and notes are Lumi’s. Pattern: Asked most years.

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