Get the Application of Integrals Class 12 NCERT Solutions for Class 12 Mathematics Chapter 8 Application of Integrals Miscellaneous Exercise below. The the PDF 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 6 marks in the Class 12 board paper, typically one long-answer modulus or trigonometric area question plus a 1-mark MCQ distractor.
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.
Why the Miscellaneous Exercise of Class 12 Maths Chapter 8 Matters Most
The this chapter address this in the same order as the NCERT textbook.
Exercise 8.1 rewards a memorised ellipse formula. The Miscellaneous Exercise refuses to. Every question here hides a sign change, and the marking scheme gives zero credit for the signed integral when the question asks for geometric area. The single most common reason students lose the full 5-mark area question in CBSE is forgetting the modulus on a curve that crosses the x-axis. This exercise is the drill that closes that gap before the board exam and JEE Main 2026.
Three of the five problems (Q3 on sin x , Q4 on x3 , Q5 on x|x| ) are explicitly designed so that the signed integral gives a tempting wrong answer, usually zero or a negative number. Recognising the trap at sight is worth more in the exam than the integration itself.
Application of Integrals Misc Video Walkthrough
Source: Magnet Brains on YouTube
How the NCERT Solutions Class 12 Maths on the Application of Integrals Class 12 NCERT Solutions Help You
Each solution in these notes is written so the sign decision comes first, before any antiderivative is applied. The Collegedunia faculty mark the zero of the integrand, split the interval there, and only then integrate, which is the exact sequence CBSE markers expect.
- Sign change located first: every modulus or odd-function solution names the split point before integrating.
- Geometric cross-check: Q2 (the |x+3| V-graph) is verified as two congruent triangles, area 92 each.
- Distractor decoded: Q4 and Q5 show which option is the "forgot the modulus" trap, so you can eliminate it by inspection.
- Symmetry shortcuts: the odd-function rule -aa|f|=20a|f| is applied where it saves a step.
Class 12 Maths Chapter 8 Miscellaneous Exercise: The Sign Rule You Apply
The this Class 12 page address this in the same order as the NCERT textbook.
One rule governs the whole exercise. Read it before opening the NCERT Solutions Class 12 Maths PDF.
Geometric area: A=ab |f(x)| dx . Wherever f changes sign at an interior point c , split: A=|ac f|+|cb f| . The plain integral ab f dx is the signed value, not the area.
The supporting antiderivatives are the power rule ∫ xn dx=xn+1n+1+C for n≠ -1 , and the trigonometric pair x dx=-cos x+C , x dx=sin x+C . For a modulus integrand, split at the zero of the inside expression, exactly as |x+3| splits at x=-3 in Q2.
Question-Wise Answer Map for Class 12 Maths Chapter 8 Miscellaneous Exercise
The the resource address this in the same order as the NCERT textbook.
The table below lists each Miscellaneous question, its curve, and the verified final value. Use it as a checking grid after attempting the set.
| Q No. | Curve and limits | Key idea | Answer |
|---|---|---|---|
| 1(i) | y=x2 , x=1 to x=2 , x-axis | Power rule, curve above axis | 73 sq units |
| 1(ii) | y=x4 , x=1 to x=5 , x-axis | Power rule, curve above axis | 31245 sq units |
| 2 | y=|x+3| , -60|x+3| dx | Split modulus at x=-3 | 9 sq units |
| 3 | y=sin x , x=0 to x=2π | Split at x=π , two arches | 4 sq units |
| 4 (MCQ) | y=x3 , x-axis, x=-2 to x=1 | Split at x=0 , add absolute values | 174 [option (D)] |
| 5 (MCQ) | y=x|x| , x-axis, x=-1 to x=1 | Odd function, 201x2 dx | 23 [option (C)] |
Q1 has no sign change because even powers stay above the axis. Q2 to Q5 all need a split. In Q4 the trap option is the signed value -154 , and in Q5 it is the signed value 0, the classic "forgot the modulus" distractors.
Step-by-Step Method Used in the NCERT Solutions for the Miscellaneous Exercise
Every solved problem in this PDF runs the same four-step routine, which keeps the sign handling examination-ready.
- Sketch the curve on the given interval and mark where it crosses the x-axis.
- Split the integral at every interior zero of the integrand (or of the inside expression for a modulus).
- Integrate each piece with the power rule or the trigonometric antiderivatives, then take absolute values.
- Add and cross-check with geometry (triangle or rectangle bound) or with an odd/even symmetry argument.
The geometric cross-check is the fastest way to catch a sign slip. For Q2 the answer 9 is two triangles of area 92 ; for Q3 the answer 4 is two sine arches of area 2 each.
Solved Example from the Miscellaneous Exercise: Area Under y=sin x in Q3
Question 3 asks for the area bounded by y=sin x between x=0 and x=2π . The curve is positive on [0,π] and negative on [π,2π] , so the modulus split is mandatory.
Sample step: First arch, A1=0πsin x dx=[-cos x]0π=1+1=2 .
Sample step: Second arch, A2=π2π(-sin x) dx=[cos x]π2π=1+1=2 . Total A=A1+A2=4 . The plain integral 02πsin x dx=0 is the trap.
Common Mistakes Students Make in Class 12 Maths Chapter 8 Miscellaneous Exercise
The chapter notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Not splitting the modulus |x+3| at x=-3 in Q2, which collapses the answer to the signed value 0.
- Computing 02πsin x dx and writing 0 as the area in Q3 instead of using |sin x| .
- Picking option (A) in Q5 because the signed integral of the odd function x|x| over [-1,1] is 0.
- Treating y=x4 as if it might dip below the axis. Even powers stay non-negative, so no modulus is needed in Q1(ii).
- Forgetting the geometric cross-check that would have caught the sign slip immediately.
Application of Integrals Class 12 NCERT Solutions: All Exercises
Chapter 8 pairs one core exercise with this consolidating set. Use the table below to revisit the ellipse and circle templates before attempting the mixed sign-handling problems.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 8.1 | Exercise 8.1 Solutions (Area of Ellipse, Circle and Parabola) |
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
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
the PDF: 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 Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 8 Application of Integrals Misc 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 under the given curves and given lines:
(i) y=x2, x=1, x=2 and the x-axis.
(ii) y=x4, x=1, x=5 and the x-axis.
Sketch the graph of y=|x+3| and evaluate -60|x+3| dx.
Find the area bounded by the curve y=sin x between x=0 and x=2π.
Area bounded by the curve y=x3, the x-axis and the ordinates x=-2 and x=1 is
(A) -9 (B) -154 (C) 154 (D) 174
The area bounded by the curve y=x |x|, the x-axis and the ordinates x=-1 and x=1 is given by
(A) 0 (B) 13 (C) 23 (D) 43
[Hint: y=x2 if x>0 and y=-x2 if x<0.]
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 the Class 12 Maths Chapter 8 Miscellaneous Exercise?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 8 Application of Integrals has 5 questions in the 2026-27 NCERT. Q1 has two power-curve parts, Q2 is a modulus integral, Q3 is a sine area over a full period, and Q4 and Q5 are single-correct MCQs on sign-changing curves.
Ques. Why is the area under y =sin x from 0 to 2π equal to 4 and not 0?
Ans. The signed integral 02πsin x dx is 0 because the positive arch on [0,π] and the negative arch on [π,2π] cancel. Geometric area uses |sin x| , so each arch contributes 2 and the total area is 4 square units. This is Q3 of the Miscellaneous Exercise.
Ques. How do you evaluate -6 0 | x +3| dx in the Miscellaneous Exercise?
Ans. Split at the zero of x+3 , which is x=-3 . On [-6,-3] , |x+3|=-(x+3) and the piece is 92 ; on [-3,0] , |x+3|=x+3 and the piece is also 92 . The sum is 9 square units, matching two triangles of area 92 each.
Ques. What is the area bounded by y =x 3 between x =-2 and x =1 ?
Ans. The cubic is below the axis on [-2,0] and above on [0,1] . Using |x3| , the left piece is 4 and the right piece is 14 , so the area is 174 square units, option (D). The signed value -154 is the trap option.
Ques. How do I download the Class 12 Maths Chapter 8 Miscellaneous Exercise NCERT Solutions PDF?
Ans. Use the download button on the NCERT Solutions Class 12 Maths card at the top of these notes to save the Collegedunia Class 12 Maths Chapter 8 Application of Integrals Miscellaneous Exercise NCERT Solutions PDF. The file is free, ad-free, and mapped to the 2026-27 NCERT edition.
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