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.
Use the resource above alongside the chapter breakdown below.
Use the resource above alongside the chapter breakdown below.
Use the resource above alongside the chapter breakdown below.
Use the resource above alongside the chapter breakdown below.
Use the resource above alongside the chapter breakdown below.
- 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.
Use the resource above alongside the chapter breakdown below.
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
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.
Other 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 8 | Application of Integrals 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.
Student Feedback
In a poll of 1,200 Class 12 students, 78% said this Application of Integrals formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.
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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