The Application of Integrals Class 12 Formulas hosted on the Application of Integrals Class 12 Formulas condense Class 12 Mathematics Chapter 8 Application of Integrals into a single ready-reference document. The formula sheet list each formula in its canonical form and pair it with a solved examples identifier from the NCERT textbook.
- CBSE Weightage: 5 to 6 marks (one 5-mark long-answer area question in nearly every paper).
- JEE Main Weightage: 2 to 3% of the Maths section (1 to 2 questions per paper, usually area between two curves).
- CUET (UG) Weightage: 1 to 2 MCQs across most shifts.
You can find the complete Formula Sheet for Application of Integrals with the area-under-curve learn formula, area-between-curves template, standard-region results for circles, ellipses and parabolas, and the modulus rule below.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE and JEE Main papers.
Application of Integrals Chapter Weightage in Class 12 Maths Across CBSE, JEE Main and CUET
The table below compares the marks share of Application of Integrals across the three exams Class 12 students sit. JEE Main 2025 led the year with two questions tagged to the Application of Integrals Class 12 Formulas.
| Exam | 2025 | 2024 | 2023 | 2022 | 2021 |
|---|---|---|---|---|---|
| JEE Main (Maths) | 2 Qs | 1 Q | 2 Qs | 1 Q | 1 Q |
| CBSE Class 12 Board | 5 marks | 6 marks | 5 marks | 5 marks | 6 marks |
| CUET (UG) Maths | 2 Qs | 1 Q | 2 Qs | 1 Q | - |
Application of Integrals is the most predictable scoring block in Class 12 Maths: the long-answer pattern repeats almost identically every year, only the curve changes.
Application of Integrals Video Walkthrough
Source: Magnet Brains on YouTube
Area Under a Curve learn Formula for Class 12 Maths Chapter 8
The Application of Integrals Class 12 Formulas address this in the same order as the NCERT textbook.
NCERT Section 8.2 introduces the area-under-curve definite integral as the limit of a Riemann sum. The four templates below cover every CBSE-tested orientation: curve above or below the x-axis, curve to the right or left of the y-axis.
| Region | Area Formula | NCERT Section | Sign Convention |
|---|---|---|---|
| Between y = f(x) , the x-axis, and x = a, x = b where f(x) ≥ 0 | A = ab f(x) dx | 8.2 | Positive integrand |
| Between y = f(x) , the x-axis, and x = a, x = b where f(x) ≤ 0 | A = | ab f(x) dx | | 8.2 | Take modulus |
| Between x = g(y) , the y-axis, and y = c, y = d where g(y) ≥ 0 | A = cd g(y) dy | 8.2 | Integrate w.r.t. y |
| Curve crosses the x-axis between a and b at x = c | A = | ac f(x) dx | + | cb f(x) dx | | 8.2 | Split at zero of f |
Area Between Two Curves Formula and the Top-Minus-Bottom Rule
The Application of Integrals Class 12 Formulas address this in the same order as the NCERT textbook.
NCERT Section 8.3 covers the area enclosed between two curves. The single most-tested CBSE 5-mark pattern asks for the area between a parabola and a line, or between two parabolas; the formula below handles every variant.
The intersection abscissae a, b and any internal crossing c come from solving f(x) = g(x) . CBSE awards 1 mark for the correct intersection points alone.
Standard Region Area Formulas for Conics in Class 12 Maths Chapter 8
The Application of Integrals Class 12 Formulas address this in the same order as the NCERT textbook.
The conic-section results below are derived in NCERT Section 8.2 examples and exercises. Memorising the closed-form area saves three to four minutes of integration time inside the CBSE board paper.
| Region | Equation | Closed-Form Area | PYQ Hits (last 5 years) |
|---|---|---|---|
| Full circle | x2 + y2 = a2 | π a2 | CBSE 2022 |
| First-quadrant quarter of circle | x2 + y2 = a2, x, y ≥ 0 | π a24 | CBSE 2023, 2025 |
| Full ellipse | x2a2 + y2b2 = 1 | π a b | CBSE 2024 |
| First-quadrant quarter of ellipse | x2a2 + y2b2 = 1, x, y ≥ 0 | π a b4 | CBSE 2021, JEE Main 2024 |
| Parabola y2 = 4ax and its latus rectum x = a | y2 = 4ax, x = a | 8 a23 | CBSE 2022, 2025 |
| Parabola x2 = 4ay and line y = a | x2 = 4ay, y = a | 8 a23 | JEE Main 2023 |
| Region inside circle x2 + y2 = a2 and above the line y = b for 0 ≤ b ≤ a | Circular segment | a2 cos-1ba - b√a2 - b2 | JEE Main 2025 |
| Triangle with vertices (x1, y1), (x2, y2), (x3, y3) via integration | Line segments | 12| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) | | CBSE 2024 |
The first-quadrant ellipse quarter and the parabola-latus-rectum region together account for over half of all Application of Integrals questions in CBSE board papers since 2019.
Sign Rule and Modulus Convention for Area Computation
The single most common source of mark loss is treating a signed integral as a geometric area. Area is always non-negative; apply the modulus on each piece when f changes sign, then add. Never sum the signed integrals first and then take the modulus.
How will Collegedunia's Application of Integrals Formula Sheet help You?
The Class 12 Maths Chapter 8 formula sheet from Collegedunia is built for the final 24-hour revision window. It compresses the Application of Integrals Class 12 Formulas into eight pages mapped to NCERT 8.2 and 8.3 so you can recognise the question type within ten seconds.
- Section-mapped layout mirrors NCERT 8.2 (area under a curve) and 8.3 (area between curves) for one-step textbook cross-reference.
- Closed-form area column lists every standard-region result (circle, ellipse, parabola) so you avoid re-deriving them in the exam hall.
- Sign-rule callout highlights the modulus convention that costs students a 5-mark question every year.
- Compact 8-page PDF prints cleanly on four A4 sheets (double-sided) for night-before revision.
Most-Asked Application of Integrals Formulas from CBSE Class 12 Board Papers
Below are the top five Application of Integrals formulas ranked by CBSE board-paper frequency across 2021 to 2025. Each has been tested at least twice in the last five years.
- A = ab [f(x) - g(x)] dx (area between two curves): tested in CBSE 2021, 2023, 2024, 2025.
- Quarter-ellipse area π a b4 : tested in CBSE 2021, 2024.
- Parabola-and-latus-rectum area 8 a23 : tested in CBSE 2022, 2025.
- Quarter-circle area π a24 : tested in CBSE 2023, 2025.
- Split-interval modulus rule |ac f| + |cb f| for sign-changing f: tested in CBSE 2022, 2024.
Full year-wise PYQ map: Class 12 Maths Chapter 8 NCERT Solutions.
Related Resources for Class 12 Maths Chapter 8 Application of Integrals
- NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals
- NCERT Notes for Class 12 Maths Chapter 8 Application of Integrals
- NCERT Book PDF for Class 12 Maths Chapter 8 Application of Integrals
- NCERT Exemplar Solutions for Class 12 Maths Chapter 8 Application of Integrals
- Handwritten Notes for Class 12 Maths Chapter 8 Application of Integrals
NCERT Formula Sheet for Class 12 Maths: All Chapters
The table below lists every Class 12 Maths chapter formula sheet currently published on Collegedunia, mapped to the 2026-27 NCERT.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Application of Integrals Class 12 Formulas: 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 application of integrals class 12 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 application of integrals class 12 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 application of integrals class 12 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.
- the PDF previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The this chapter 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 this Class 12 page 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 the resource 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 chapter notes 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 PDF ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The application of integrals class 12 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 Class 12 Application of Integrals | 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 application of integrals class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board the chapter notes 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 the PDF 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 the this chapter 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 these notes 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.
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.
Application of Integrals Class 12 Formulas - Frequently Asked Questions
Ques. What is the formula for area under a curve in Class 12 Maths Chapter 8?
Ans. The area bounded by the curve y = f(x) , the x-axis, and the vertical lines x = a and x = b is given by A = ab f(x) dx when f(x) ≥ 0 on [a, b]. When f(x) ≤ 0 on the interval, take the modulus of the integral.
Ques. How do you find the area between two curves in Class 12 Maths?
Ans. If f(x) ≥ g(x) on [a, b], the area enclosed between the curves y = f(x) and y = g(x) is A = ab [f(x) - g(x)] dx .
The limits a and b are found by solving f(x) = g(x) . If the curves cross inside the interval, split at the crossing point and add the absolute values.
Ques. What is the weightage of Application of Integrals in Class 12 Maths CBSE board paper?
Ans. Application of Integrals carries 5 to 6 marks in the CBSE Class 12 Maths board paper, almost always as a single 5-mark long-answer question that asks for the area of a region bounded by two curves or by a conic and a line.
Ques. What is the area of a quarter-circle of radius a in Class 12 Maths?
Ans. The first-quadrant quarter of the circle x2 + y2 = a2 has area π a24 . The derivation uses A = 0a √a2 - x2 dx and the standard antiderivative ∫ √a2 - x2 dx = x2√a2 - x2 + a22sin-1xa + C from Chapter 7.
Ques. What is the area enclosed by an ellipse in Class 12 Maths?
Ans. The area enclosed by the ellipse x2a2 + y2b2 = 1 is π a b . The first-quadrant quarter has area π a b4 . The result generalises the circle formula π a2 (obtained when a = b ).
Ques. How do I download the Class 12 Maths Chapter 8 Application of Integrals formula sheet PDF?
Ans. Use the green download button on the Application of Integrals Class 12 Formulas card at the top of the Application of Integrals Class 12 Formulas to save the 8-page this resource Class 12 Maths Chapter 8 Application of Integrals formula sheet to your device. the Application of Integrals Class 12 Formulas is free, ad-free, and mapped to the 2026-27 NCERT edition.
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