The NCERT Exemplar Class 12 Maths Integrals PDF provided here include the complete NCERT Exemplar chapter for Class 12 Mathematics Chapter 8 Application of Integrals. The NCERT Exemplar Class 12 Maths Integrals PDF are aligned to the 2026-27 NCERT release and serve as the standard practice set after the main textbook exercises.
- CBSE Weightage: 5-6 marks (typically one 5-mark LA on area between two curves)
- JEE Main Weightage: 2-3% (1 question per shift, almost always a single-correct MCQ)
- JEE Main Weightage: Not applicable (Mathematics is not a JEE Main subject)
Student Pulse - Application of Integrals Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
The chapter is short on theory: four formulas in section 8.1, then straight to problems. The page below carries the problem-range map, question-type split, Exemplar vs textbook step-up and a parallel-track plan.
This Exemplar PDF is sourced from the official NCERT print and tagged by curve-pair for drilling one family at a time.
Also Check:
- Application of Integrals Class 12 Maths NCERT Exemplar Solutions
- Application of Integrals Class 12 Maths NCERT Book PDF
Why the NCERT Exemplar Matters for Class 12 Maths Application of Integrals
The NCERT Exemplar Class 12 Maths Integrals PDF address this in the same order as the NCERT textbook.
The textbook carries 17 problems plus a short Miscellaneous Exercise. The Exemplar adds 34 more across every curve-pair JEE Main and CBSE recycle.
- Curve-Pair Variety: Parabola-line, parabola-parabola, ellipse, circle-line, modulus and parametric ( x = at2, y = 2at ). The textbook covers fewer than half.
- Sketch-Then-Integrate Habit: Over 70% of Exemplar problems explicitly demand a sketch. CBSE awards 1 mark for it on every 5-mark LA.
- JEE-Pattern MCQs: 11 Exercise MCQs plus 3 Solved-Example MCQs mirror the JEE Main single-correct format.
Application of Integrals NCERT Exemplar Video Walkthrough
Source: Shivang Maths Academy on YouTube
How the NCERT Exemplar Class 12 Maths Integrals PDF on the NCERT Exemplar Class 12 Maths Integrals PDF Help You
The NCERT Exemplar Class 12 Maths Integrals PDF address this in the same order as the NCERT textbook.
The bare Exemplar print is a problem book with almost no scaffolding. This Collegedunia download adds prep aids for the JEE Main and CBSE Board double track.
- Four-Formula Reference Card: The four area formulas in section 8.1, including the split-integral case.
- Curve-Pair Tagging: Each problem tagged by curve pair so you can drill one family at a time.
- Question-Type Tagging: SA, LA or MCQ label on every problem.
- Sketch Hint Annotations: Each LA carries the sketch shape to draw before integrating.
Application of Integrals Exemplar Question-Type Distribution
The NCERT Exemplar Class 12 Maths Integrals PDF address this in the same order as the NCERT textbook.
The 48 problems split across three types. CBSE-prep students prioritise SA and LA; JEE aspirants prioritise MCQs plus Ex 10-12.
| Type | Count | Marks Each | Total Marks | Best For |
|---|---|---|---|---|
| Solved Examples | 14 | - | - | Skill modelling |
| Short Answer (SA) | 15 | 2-4 | ~45 | CBSE SA Section |
| Long Answer (LA) | 8 | 5-6 | ~44 | CBSE 5-mark LA |
| MCQ (Objective Type) | 11 | 1 | 11 | JEE Main + CBSE MCQ |
| Fill-in (Solved Examples) | 2 | - | - | Recall drills |
What's Inside the NCERT Exemplar Class 12 Maths Integrals PDF
Every entry in the NCERT Exemplar Class 12 Maths Integrals PDF is reproduced in the order it appears in the NCERT textbook.
The 9 pages run as one connected problem set with minimal theory. The table below maps each problem range to its type and pages.
| Section | Problem Range | Pages | What It Covers |
|---|---|---|---|
| 8.1 Overview (formulas) | 4 formulas | 1 | Four area formulas plus split-integral case |
| Solved Examples: SA + LA | Ex 1 to Ex 9 | 1-6 | Sine, ay2=x3 , parabola-line, two parabolas, ellipse, cycloid, circle segment |
| Solved Examples: MCQ + Fill-in | Ex 10 to Ex 14 | 6-7 | Area MCQs on circle, ellipse, y=x2 ; recall fill-ins |
| Exercise: Short Answer | Q1 to Q15 | 7-8 | Parabola-line, two-parabolas, cubic-line, semicircle, modulus |
| Exercise: Long Answer | Q16 to Q23 | 8 | Parabola-circle, sine on [0,2π] , triangle, inequality region, modulus |
| Exercise: Objective (MCQ) | Q24 to Q34 | 8-9 | Single-correct MCQs on circle, ellipse, parabola, sine, modulus |
Exemplar vs NCERT Textbook Difficulty Comparison for Application of Integrals
The textbook tests one curve at a time and gives the sketch. The Exemplar pairs two curves and uses parametric or implicit forms.
| Concept | NCERT Textbook Style | NCERT Exemplar Style |
|---|---|---|
| Area Under One Curve | "Find the area bounded by y = x2 , x-axis, x = 1 , x = 4 ." | "Find the area bounded by ay2 = x3 , the y-axis and y = a, y = 2a ." |
| Area Between Two Curves | "Find the area between y = x2 and y = x ." | "Find the area bounded by y2 = 6x and x2 = 6y ; verify using both ∫ y dx and ∫ x dy ." |
| Ellipse / Parametric | "Find the area of x2a2 + y2b2 = 1 ." | "Find the area enclosed by x = 3cos t, y = 2sin t using the parametric form." |
| Circle Segment | Not in main exercises. | "Find the area of the minor segment of x2 + y2 = a2 cut by x = a/2 ." |
The Exemplar always asks you to find intersection points, sketch the region, then integrate. Textbook-only students lose 1-2 marks per CBSE 5-mark LA because Boards now copy this layered style.
Full learn sheet: this Class 12 page Maths NCERT Exemplar Solutions
JEE Main Prep Value of the Application of Integrals Exemplar PDF
Application of Integrals appears in nearly every JEE Main shift as a single-correct MCQ on area enclosed by standard curves. Exemplar MCQs are calibrated to this pattern.
| JEE Main Question Pattern | Matching Exemplar Problem | Why It Trains the Pattern |
|---|---|---|
| Area enclosed by a circle | Q32, Ex 10 | Direct π r2 plus integral verification |
| Area enclosed by an ellipse | Q31, Ex 11 | π ab derivation from ∫ 4 ba√a2 - x2 dx |
| Area between parabola and line | Q25, Q29, Ex 3 | Intersection-first, sign-aware set-up |
| Area under a trigonometric curve | Q24, Q28, Q30, Ex 1 | Modulus handling for sin x, cos x on [0, 2π] |
| Region defined by an inequality | Q19 ( y2 ≤ 6ax, x2 + y2 ≤ 16a2 ) | Almost identical to JEE Main 2024 Shift 2 |
Old vs Current 2026-27 Syllabus Alignment for Chapter 8 Exemplar
The Exemplar was not updated during the 2022-23 revision, but the Chapter 8 textbook itself changed very little, so the Exemplar remains almost fully aligned with the current 2026-27 syllabus.
| Topic | In 2026-27 Textbook? | In Exemplar PDF? | Should You Solve? |
|---|---|---|---|
| Area under y = f(x) | Yes | Yes | Priority 1 |
| Area between two curves | Yes | Yes | Priority 1 |
| Area of ellipse and circle | Yes | Yes | Yes |
| Parametric form (Ex 5, Ex 7) | Trimmed | Yes | Useful for JEE; skip if pure CBSE |
| Region by inequalities (Q19) | Yes | Yes | Yes, esp. for JEE |
Roughly 5% of Exemplar problems use parametric forms the current textbook only shows as solved examples. JEE aspirants should solve them all; board-only students can skip Ex 5 and Q7 if pressed.
Detailed kept-trimmed-removed table: this chapter Maths NCERT Book PDF
How to Use the Exemplar with the NCERT Textbook: 4-Day Parallel-Track Plan
The Exemplar is not a replacement for the NCERT Exemplar Class 12 Maths Integrals PDF. Interleave them: read, then attempt Exemplar problems on the same topic.
- Day 1 (Area Under One Curve): Read Section 8.1 and first half of 8.2; attempt Exemplar SA Q1, Q7, Q9, Q10, Q12 and Solved Examples 1-2.
- Day 2 (Area Between Two Curves): Finish Section 8.2; attempt Exemplar SA Q2-Q6, Q14-Q15 under a 60-minute timer.
- Day 3 (Standard Curves): Revise standard-form integrals; attempt LA Q16, Q19, Q22 plus Ex 4, 8, 9 in 75 minutes.
- Day 4 (Objective Drill): MCQs Q24-Q34 plus Ex 10-12 back to back at 90 seconds per MCQ for JEE Main calibration.
Related Links:
More Application of Integrals Class 12 Maths Resources
NCERT Exemplar Book PDF for Class 12 Mathematics: All Chapters
Download any other chapter's Exemplar PDF below.
| Chapter | Exemplar Book PDF |
|---|---|
| Chapter 1 | Relations and Functions NCERT Exemplar Book PDF |
| Chapter 2 | Inverse Trigonometric Functions NCERT Exemplar Book PDF |
| Chapter 3 | Matrices NCERT Exemplar Book PDF |
| Chapter 4 | Determinants NCERT Exemplar Book PDF |
| Chapter 5 | Continuity and Differentiability NCERT Exemplar Book PDF |
| Chapter 6 | Application of Derivatives NCERT Exemplar Book PDF |
| Chapter 7 | Integrals NCERT Exemplar Book PDF |
| Chapter 9 | Differential Equations NCERT Exemplar Book PDF |
| Chapter 10 | Vector Algebra NCERT Exemplar Book PDF |
| Chapter 11 | Three Dimensional Geometry NCERT Exemplar Book PDF |
| Chapter 12 | Linear Programming NCERT Exemplar Book PDF |
| Chapter 13 | Probability NCERT Exemplar Book PDF |
NCERT Exemplar Class 12 Maths Integrals PDF: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Application of Integrals Chapter
The Application of Integrals chapter splits into 1 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 8.1 | Area under simple curves and between two curves |
| Miscellaneous Exercise | Mixed application of integrals problems |
PDF Download Formats and Languages for the Application of Integrals Chapter
The Application of Integrals Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The the resource ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the chapter notes ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The application of integrals class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The application of integrals class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The the PDF solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this application of integrals class 12 pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the application of integrals class 12 formulas pdf), and the full application of integrals class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Application of Integrals Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Application of Integrals chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable application of integrals class 12 important questions you will see on board day.
- this chapter previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The application of integrals class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching these notes extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Application of Integrals chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full application of integrals class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Application of Integrals Notes Pair with NCERT Solutions and the Formula Sheet
The this Class 12 page notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Application of Integrals Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| the resource ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| application of integrals class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The the chapter notes ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The the PDF solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The application of integrals class 12 formulas reference sheet is the same A4 file students sometimes refer to as this chapter all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board these notes textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 application of integrals solutions, class 12 application of integrals ncert solutions, ncert class 12 application of integrals solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to this Class 12 page |
|---|---|
| RD Sharma the PDF | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal this chapter | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the resource | Free online walkthroughs; useful for video-style learning. |
| Shaalaa the chapter notes solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board the PDF textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar these notes | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Application of Integrals Notes Page Most Effectively
The recommended study plan for the this chapter chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use these notes ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The application of integrals class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
This Collegedunia NCERT Class 12 Mathematics page is reviewed against every CBSE board paper release.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
NCERT Exemplar Class 12 Maths Integrals PDF - Frequently Asked Questions
Ques. Where can I download the NCERT Exemplar Class 12 Maths Integrals PDF for free?
Ans. You can download these notes Maths NCERT Exemplar Book PDF directly from the NCERT Exemplar Class 12 Maths Integrals PDF. Both Normal and HD versions are free.
Ques. How many pages and how many problems are in the Class 12th Maths Application of Integrals Exemplar PDF?
Ans. The Exemplar PDF for Chapter 8 runs 9 pages. It contains 14 solved examples plus 34 unsolved problems split as 15 SA, 8 LA and 11 MCQ (Objective Type).
Ques. Is this NCERT Exemplar Class 12 Maths Integrals PDF aligned with the 2026-27 NCERT syllabus?
Ans. Yes, almost fully. The Exemplar itself was not revised during the 2022-23 update, but the Chapter 8 textbook itself changed very little. Only the parametric-form Solved Examples (Ex 5 and Q7) sit slightly beyond the strict CBSE-only scope; JEE aspirants should still attempt them.
Ques. Is the NCERT Exemplar harder than the NCERT textbook for Application of Integrals?
Ans. Yes. The Exemplar pairs two curves in almost every problem, demands a rough sketch and intersection points before the integral, and uses parametric or implicit forms more often. This is the style that CBSE Boards now use for 5-mark LA on the NCERT Exemplar Class 12 Maths Integrals PDF, and the format JEE Main uses for its single-correct MCQ.
Ques. How important is the NCERT Exemplar for JEE Main preparation on Application of Integrals?
Ans. Very important. JEE Main carries 1 question on Application of Integrals in almost every shift, and the question is calibrated to the Exemplar style: area enclosed by a standard curve pair or a region defined by an inequality. Exemplar Q19 ( y2 ≤ 6ax and x2 + y2 ≤ 16a2 ) is almost identical to a JEE Main 2024 Shift 2 problem.
Ques. Which Exemplar problems most resemble the CBSE Board long-answer style for Application of Integrals?
Ans. Long Answer problems Q16 (parabola-circle), Q19 (region inequality) and Q22 (modulus of 2cos x ) most closely match CBSE Board 5-mark style. Solved Examples 6, 8 and 9 are also at exactly this difficulty.
Ques. Should I attempt the Exemplar before or after the NCERT textbook for Chapter 8?
Ans. Attempt them in parallel. Read a topic from the NCERT Exemplar Class 12 Maths Integrals PDF, solve the matching textbook exercise, then attempt Exemplar problems on the same topic. Trying the Exemplar before the NCERT Exemplar Class 12 Maths Integrals PDF usually leaves students stuck on intersection points and sketch construction.
Ques. What is the formula for area enclosed between two curves used across the Exemplar?
Ans. For curves y = f(x) and y = g(x) with f(x) ≥ g(x) on [a, b] , the area is ab [f(x) - g(x)] dx . When the relative position flips at some c ∈ (a, b) , split the integral:
ac [f(x) - g(x)] dx + cb [g(x) - f(x)] dx . The Exemplar section 8.1 lists this as formula 8.1.4 and roughly a third of the LA problems rely on it.
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