Get the this chapter for Class 12 Mathematics Chapter 8 Application of Integrals Exercise 8.1 below. These notes answer every part of every question, observe CBSE step-marking conventions, and form part of the solutions PDF-wide NCERT Solutions Class 12 Maths set hosted on this site.
- CBSE Weightage: 4 to 5 marks in the Class 12 board paper, usually one long-answer area question worth 5 marks plus an occasional 1-mark MCQ.
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.
Solutions curated by Collegedunia Class 12 Mathematics faculty, matched to the 2026-27 NCERT print and cross-checked against the official answer key.
Application of Integrals Class 12 NCERT Solutions: What Exercise 8.1 Contains
Exercise 8.1 is short but sets every template used later in the this Class 12 page. Q1 and Q2 are full ellipse-area derivations, Q3 is a quarter-circle MCQ, and Q4 is a parabola-area MCQ that rewards horizontal strips. The table below pairs each question with the curve it presents and the verified final area, so you can self-check after attempting the exercise.
| Q No. | Region / curve | Method | Area (answer) |
|---|---|---|---|
| 1 | Ellipse x216+y29=1 | Vertical strips, four-quadrant symmetry | 12π sq units |
| 2 | Ellipse x24+y29=1 | Vertical strips, four-quadrant symmetry | 6π sq units |
| 3 (MCQ) | First-quadrant region of circle x2+y2=4 , lines x=0 , x=2 | Quarter-disk, 02√4-x2 dx | π [option (A)] |
| 4 (MCQ) | Region under y2=4x , the y -axis and y=3 | Horizontal strips, 03y24 dy | 94 [option (B)] |
Both ellipse problems reduce to the same standard integral ∫√a2-x2 dx , and both verify against the closed form π a b . An ellipse, circle, or parabola area question has appeared in 4 of the last 5 CBSE Class 12 Maths papers, most often carrying a full 5 marks.
Application of Integrals Ex 8 1 Video Walkthrough
Source: Magnet Brains on YouTube
Class 12 Maths Chapter 8 Exercise 8.1: The Two Area Formulae You Apply
Every problem in Exercise 8.1 is built from one idea: a thin strip of area, summed by integration. Fix the two strip directions before opening the the resource PDF and the entire exercise becomes mechanical.
Vertical strip: A=ab y dx for a region bounded above by y=f(x) , below by the x -axis, between x=a and x=b . Horizontal strip: A=cd x dy when the boundaries are easier to read as x=g(y) .
For the standard-form ellipse x2a2+y2b2=1 , all four quadrants enclose equal area, so the total is A=40a y dx=π a b . For the circle x2+y2=a2 , each quadrant carries 14π a2 . The single antiderivative that closes Q1, Q2 and Q3 is ∫√a2-x2 dx=x2√a2-x2+a22sin-1xa+C , derived by the substitution x=asinθ .
Step-by-Step Method Used in the NCERT Solutions for Class 12 Maths Exercise 8.1
Every solved problem in the chapter notes follows the same four-line routine, the exact structure CBSE markers reward in a 5-mark area question.
- Sketch the region and read off the bounding curves and limits. The diagram decides whether you take vertical or horizontal strips.
- Solve the curve for the strip variable ( y in terms of x , or x in terms of y ), keeping only the sign valid in the region.
- Set up the symmetric integral and pull constants out, then apply the standard antiderivative.
- Substitute the limits and verify with the closed form ( π a b for an ellipse, 14π r2 for a circular quadrant).
The verification step is what separates a full-marks answer from a near miss. CBSE expects the closed-form cross-check written out, and the Collegedunia solutions show it on every problem.
How the Application of Integrals Class 12 NCERT Solutions on the Application of Integrals Class 12 NCERT Solutions Help You
The the PDF address this in the same order as the NCERT textbook.
Exercise 8.1 looks like four routine sums, yet it is the single highest-value 5-mark question pattern in the chapter, because the ellipse derivation reappears almost verbatim in the board paper. The Collegedunia solutions write the symmetry argument in words, keep the trigonometric substitution explicit, and never skip the π a b sanity check.
- Symmetry stated, not assumed: every ellipse solution explains why the area is four times the first-quadrant piece.
- Two routes shown: Q2 carries both a vertical-strip and a horizontal-strip derivation so the equality of the two methods is visible.
- MCQ shortcut flagged: Q3 is closed in one line as a quarter-disk, 14π r2=π , before the full integral cross-check.
- Strip-direction logic: Q4 shows why horizontal strips give a single-formula integrand while vertical strips force an awkward split.
CBSE Board Exam Relevance of Class 12 Maths Chapter 8 Exercise 8.1
The this chapter address this in the same order as the NCERT textbook.
Area-by-integration is one of the most predictable 5-mark slots in the Class 12 Maths paper. The table below maps the last five CBSE sittings against the Exercise 8.1 style question that appeared.
| Year | Marks from Ex 8.1 style | Question tested |
|---|---|---|
| 2025 | 5 | Area enclosed by an ellipse using the definite integral |
| 2024 | 5 | Area of a circular region in the first quadrant |
| 2023 | 1 | MCQ on area bounded by a parabola and a line |
| 2022 | 5 | Area of a region bounded by a curve and the coordinate axes |
| 2021 | 4 | Area under a standard curve via the √a2-x2 integral |
Across the last five sittings, Exercise 8.1 style area questions have carried a steady 4 to 5 marks almost every year. This is the most efficient 5-mark return in the Application of Integrals chapter and should be practised until the ellipse derivation is automatic.
Common Mistakes Students Make in Class 12 Maths Exercise 8.1
These notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Using vertical strips on Q4. The parabola-and-line region needs horizontal strips, else the upper boundary changes at x=94 and forces a split integral.
- Forgetting the sin-1 term when applying ∫√a2-x2 dx . Without it the ellipse area comes out wrong.
- Skipping the closed-form check. CBSE expects π a b or 14π r2 written as verification.
- Dropping the factor of 4 from the four-quadrant symmetry and computing only the first-quadrant area.
- Reading the wrong root of the curve, taking the negative branch where the region lies above the axis.
Solved Example from Exercise 8.1: Area of the Ellipse in Q1
Question 1 asks for the area enclosed by x216+y29=1 . Comparing with the standard form gives a=4 , b=3 . The full solution in the this Class 12 page runs the symmetric integral; the sample steps below show the load-bearing lines.
Sample step: Solve for the first-quadrant branch, y=34√16-x2 . By symmetry, A=40434√16-x2 dx=304√16-x2 dx .
Sample step: With a=4 , 04√16-x2 dx=[x2√16-x2+8sin-1x4]04=4π . So A=3× 4π=12π , which matches π a b=π(4)(3)=12π .
Application of Integrals Class 12 NCERT Solutions: All Exercises
The the resource address this in the same order as the NCERT textbook.
Chapter 8 has one core exercise and a consolidating miscellaneous set. Use the table below to move to the mixed-method problems once the ellipse and circle templates are secure.
| Exercise | NCERT Solutions |
|---|---|
| Miscellaneous | Miscellaneous Exercise Solutions (Power Curves, Modulus and Trigonometric Areas) |
Related Resources for Class 12 Maths Chapter 8 Application of Integrals
- Formula Sheet 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
the chapter notes: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
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.
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.
- application of integrals class 12 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 application of integrals class 12 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
These notes 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 PDF 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 this chapter 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 Class 12 page all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource 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 the chapter notes |
|---|---|
| RD Sharma Class 12 Application of Integrals | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Application of Integrals | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the PDF | 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 this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Application of Integrals | Advanced problems for JEE Main/JEE Advanced preparation. |
All NCERT Solutions for Application of Integrals Ex 8.1 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 8 Application of Integrals Ex 8.1 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Find the area of the region bounded by the ellipse x216+y29=1.
Find the area of the region bounded by the ellipse x24+y29=1.
Area lying in the first quadrant and bounded by the circle x2+y2=4 and the lines x=0 and x=2 is
(A) π (B) π2 (C) π3 (D) π4
Area of the region bounded by the curve y2=4x, the y-axis and the line y=3 is
(A) 2 (B) 94 (C) 93 (D) 92
How to Use the Application of Integrals Notes Page Most Effectively
The recommended study plan for these notes 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 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.
Application of Integrals Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 12 Maths Chapter 8 Exercise 8.1?
Ans. Exercise 8.1 of Class 12 Maths Chapter 8 Application of Integrals has 4 questions in the 2026-27 NCERT. Q1 and Q2 are long-answer derivations of the area of an ellipse, Q3 is an MCQ on the area of a quarter circle, and Q4 is an MCQ on the area bounded by a parabola and a line.
Ques. What is the area of the ellipse in Exercise 8.1 Q1 of Class 12 Maths?
Ans. For the ellipse x216+y29=1 , the semi-axes are a=4 and b=3 . Using the symmetric integral A=404 y dx the area works out to 12π square units, which also follows directly from the closed form π a b=π(4)(3) .
Ques. Which strip method should I use for the parabola question in Exercise 8.1?
Ans. Use horizontal strips for Q4. The region is bounded by the parabola y2=4x , the y -axis and the line y=3 . Writing x=y24 and integrating 03y24 dy gives a single-formula integrand and the answer 94 . Vertical strips would force an awkward split integral.
Ques. Why is the area of the circle region in Exercise 8.1 Q3 equal to π ?
Ans. The circle x2+y2=4 has radius 2. The lines x=0 and x=2 together with the first-quadrant arc bound exactly one quarter of the disk. So the area is 14π r2=14π(2)2=π , which the integral 02√4-x2 dx confirms. The correct option is (A).
Ques. How do I download the Class 12 Maths Chapter 8 Exercise 8.1 NCERT Solutions PDF?
Ans. Use the download button on the this chapter card at the top of these notes to save the Collegedunia Class 12 Maths Chapter 8 Application of Integrals Exercise 8.1 NCERT Solutions PDF. The file is free, ad-free, and mapped to the 2026-27 NCERT edition.
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