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CBSE Class 12 Mathematics Chapter 8 Application of Integrals Worksheet with Answers

Application of Integrals is a high-scoring yet precision-demanding chapter in CBSE Class 12 Mathematics. This worksheet provides structured practice across all question formats seen in board exams: objective questions to test conceptual clarity, numerical problems to build computational accuracy, and case-based scenarios to develop analytical thinking. With complete worked solutions provided, students can diagnose weak areas and strengthen their command over area calculations before the 2025 board examinations.

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Key takeaways

✓Chapter 8 Application of Integrals focuses exclusively on area under curves and area between two curves using definite integration techniques
✓The 2025 CBSE Class 12 Mathematics board exam typically allocates 5-6 marks to this chapter in both terms
✓Determining correct limits of integration and proper curve sketching are the two most common error zones for students
✓Area calculations always yield positive values; use absolute value or split integrals at intersection points accordingly
✓Case-study questions now form 4 marks in board papers and frequently use real-world scenarios involving area applications
✓NCERT exemplar problems and previous years' board questions form 60-70% of the difficulty pattern in final exams

Quick Chapter Recap: Application of Integrals

The NCERT Class 12 Mathematics Chapter 8 revolves around two core applications of definite integrals. First, the area under curves uses the formula A = ∫[a to b] y dx or A = ∫[c to d] x dy, depending on whether integration is performed with respect to x or y. The choice depends on which variable simplifies the calculation. Second, the area between two curves y = f(x) and y = g(x) from x = a to x = b is given by A = ∫[a to b] |f(x) - g(x)| dx. Identifying which curve lies above the other in the given interval is critical. When curves intersect within the interval, the area must be split at intersection points to ensure positive area values. The chapter emphasises proper sketching of curves, accurate determination of limits, and splitting integrals where necessary. Understanding symmetry properties can halve computation time for even functions integrated over symmetric intervals about the origin.

Area under y = f(x) from x = a to x = b: ∫[a to b] f(x) dx (if f(x) ≥ 0)
Area enclosed by x = g(y) from y = c to y = d: ∫[c to d] g(y) dy
Area between curves f(x) and g(x): ∫[a to b] |f(x) - g(x)| dx
Always sketch curves to identify intersection points and relative positions
For curves below the x-axis, use absolute value to ensure positive area

Worksheet Instructions and Difficulty Level

This CBSE Class 12 Mathematics worksheet is designed for students who have completed Chapter 8 Application of Integrals from the NCERT textbook and wish to assess their readiness for board examinations. Difficulty level: Moderate to Advanced, mirroring the 2025 CBSE board exam pattern. Suggested time: 90 minutes under exam conditions. The worksheet comprises five distinct sections plus one case-study question, totalling 25+ problems that cover every concept from area under simple curves to complex region calculations involving multiple curves. Section A tests conceptual understanding through multiple-choice questions. Section B checks formulae recall and quick computation via fill-in-the-blanks. Section C uses true/false statements to address common misconceptions. Sections D and E build problem-solving stamina with short and long answer questions respectively, while the case study integrates real-world context. Students should attempt all questions on separate sheets, then verify answers using the detailed answer key provided at the end. Mark allocation mirrors actual board exam weightage to help students gauge time management strategies effectively.

Total marks: 40 (mirroring board exam weightage for this chapter)
Section A: 1 mark each × 6 = 6 marks
Section B: 1 mark each × 5 = 5 marks
Section C: 1 mark each × 4 = 4 marks
Section D: 3 marks each × 5 = 15 marks
Section E: 4 marks each × 2 = 8 marks
Case Study: 1+1 = 2 marks

Section A: Multiple Choice Questions (1 mark each)

These six MCQs test foundational concepts and quick analytical skills required for CBSE Class 12 Mathematics board exams. Each question has four options with exactly one correct answer. Students must not only identify the right answer but also understand why the other three options are incorrect. Common traps include sign errors when curves lie below the x-axis, incorrect limit identification when curves intersect, and confusion between dx and dy integration forms. Time allocation: 10 minutes for this section. Mark your answers clearly and avoid wild guessing; board exams do not have negative marking, but building accuracy now prevents costly errors later. These questions draw heavily from NCERT exercises 8.1 and 8.2, plus previous years' board papers from 2020-2024, ensuring you encounter the exact style and difficulty CBSE examiners prefer.

Q1. The area bounded by the curve y = x², the x-axis and the ordinates x = 1 and x = 3 is: (a) 26/3 sq units (b) 28/3 sq units (c) 8 sq units (d) 10 sq units
Q2. 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 sq units (b) π/2 sq units (c) π/4 sq units (d) 2π sq units
Q3. The area between the curves y = x and y = x² is: (a) 1/6 sq units (b) 1/3 sq units (c) 1/2 sq units (d) 2/3 sq units
Q4. The area of the region bounded by y² = 4x, y-axis and the line y = 3 is: (a) 2 sq units (b) 9/4 sq units (c) 9/2 sq units (d) 9 sq units
Q5. Area bounded by the curve y = sin x between x = 0 and x = 2π is: (a) 0 sq units (b) 2 sq units (c) 4 sq units (d) 8 sq units
Q6. The area of the region {(x, y): 0 ≤ y ≤ x² + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} is: (a) 23/6 sq units (b) 13/6 sq units (c) 7/3 sq units (d) 5/2 sq units

Section B: Fill in the Blanks (1 mark each)

This section of the Class 12 Mathematics worksheet checks your recall of key formulae, standard results, and computational accuracy under time pressure. Each blank requires a precise numerical answer or exact algebraic expression. Approximations will not earn marks. Students commonly lose marks here by forgetting to apply limits correctly or making arithmetic slips in fraction simplification. Time allocation: 8 minutes. Write answers directly; no working is needed, but rough work on the margin helps avoid careless mistakes. These five statements are drawn from standard NCERT Class 12 Mathematics solved examples and exercise problems, ensuring alignment with the board syllabus. Practice mental calculation of simple integrals like ∫x² dx or ∫sin x dx to boost speed, as board exams reward both accuracy and efficiency in such low-mark, high-frequency questions.

Q7. The area enclosed by the ellipse x²/a² + y²/b² = 1 is __________ sq units.
Q8. The area bounded by the parabola y² = 4ax and its latus rectum is __________ sq units.
Q9. Area of the region bounded by the curve y = cos x between x = 0 and x = π is __________ sq units.
Q10. The area lying above the x-axis and included between the circle x² + y² = 8x and the parabola y² = 4x is __________ sq units.
Q11. If the area above the x-axis, bounded by the curves y = 2^(kx) and x = 0 and x = 2 is (3/log 2) sq units, then k = __________.

Section C: True or False (1 mark each)

This section targets common misconceptions and conceptual pitfalls in Application of Integrals. Students must write 'True' or 'False' for each statement and, where possible, provide a one-line justification in the answer key to reinforce understanding. CBSE examiners often test boundary cases: curves that touch but do not cross, integrals over symmetric intervals, and situations where area formulas must be modified. Time allocation: 6 minutes. Hasty reading causes most errors here; for instance, confusing 'area between curves' with 'area under a single curve,' or ignoring absolute value requirements when a curve dips below the x-axis. These four statements are modelled on assertion-reason questions and common board exam traps from the 2022, 2023, and 2024 CBSE Class 12 Mathematics papers, giving you authentic exam exposure and building critical reasoning skills essential for tackling unfamiliar problem variations.

Q12. The area bounded by the curve y = f(x), x-axis and two ordinates x = a and x = b can be negative.
Q13. For calculating area between two curves, it is always necessary to find their points of intersection.
Q14. If a curve is symmetrical about the y-axis, the area from x = -a to x = a is twice the area from x = 0 to x = a.
Q15. The area enclosed by |x| + |y| = 1 is 2 sq units.

Section D: Short Answer Questions (3 marks each)

Section D contains five short-answer problems, each worth 3 marks, designed to build procedural fluency and step-by-step reasoning. A complete answer typically requires: (i) a rough sketch of the curve(s), (ii) identification of limits and setup of the integral, and (iii) evaluation with proper substitution or use of standard integral results from NCERT Class 12 Mathematics. Partial marks are awarded for correct setup even if final computation has minor errors, so always show your working clearly. Time allocation: 25 minutes, approximately 5 minutes per question. These questions are sourced from NCERT exercises 8.1, 8.2, and exemplar problems, ensuring direct relevance to board exam patterns. Practice writing neatly; examiners in board exams award step-wise marks, and illegible work loses credit. Common mistakes include forgetting to split the integral at intersection points, sign errors when integrating below the x-axis, and incorrect limit substitution for curves expressed in parametric or polar form, though this chapter primarily uses Cartesian coordinates.

Q16. Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis in the first quadrant.
Q17. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32.
Q18. Find the area bounded by the parabola y = x² and the line y = 4.
Q19. Using integration, find the area of the region bounded by the triangle whose vertices are (1,0), (2,2) and (3,1).
Q20. Sketch the graph of y = |x + 3| and evaluate ∫[-6 to 0] |x + 3| dx. Hence, find the area bounded by the curve and the x-axis between x = -6 and x = 0.

Section E: Long Answer and HOTS Questions (4 marks each)

Section E presents two long-answer or Higher Order Thinking Skills (HOTS) questions, each carrying 4 marks and demanding multi-step reasoning, integration of concepts, or real-world application. These mirror the complexity of 4-mark questions in the 2025 CBSE board exam and test whether you can synthesise ideas rather than just reproduce standard methods. Time allocation: 15 minutes, about 7-8 minutes per question. A full-credit answer must include a clear diagram, mathematical setup with reasoning, correct integral evaluation, and a final answer with units. Examiners look for logical flow and justification of each step. HOTS questions may involve finding area when curves are given implicitly, optimising area under constraints, or splitting regions based on inequalities. Class 12 Mathematics solutions manuals and NCERT exemplar problems are excellent resources for similar questions. Always double-check final answers for dimensional consistency (square units) and reasonableness; an area of 1000 sq units for simple polynomial curves should raise a red flag prompting recalculation.

Q21. Find the area of the smaller region bounded by the ellipse x²/9 + y²/4 = 1 and the line x/3 + y/2 = 1.
Q22. Using integration, find the area of the region bounded by the parabola y² = 4x and the circle 4x² + 4y² = 9.

Case Study Question (1+1 = 2 marks)

Case-study questions are a recent addition to the CBSE Class 12 Mathematics board exam pattern, introduced to assess application skills and real-world problem-solving. This 2-mark case typically presents a contextual scenario—such as land area calculation, design of engineering components, or ecological survey regions—followed by two sub-questions of 1 mark each. Students must extract relevant mathematical data from the narrative, model it using curves or regions, set up appropriate integrals, and compute area. Time allocation: 8 minutes. Board examiners favour scenarios from agriculture (irrigated plots bounded by canals), architecture (window designs using curves), and environmental science (forest cover estimation). NCERT Class 12 Mathematics does not have in-text case studies for Chapter 8, but sample papers from CBSE and previous board sets (2023 Term 2, 2024 annual) offer excellent practice. Read the scenario twice, underline numerical data and curve equations, sketch the region, and only then attempt the sub-questions to avoid misinterpretation that costs both marks.

Complete Answer Key with Explanations

This detailed answer key provides step-by-step solutions for every question in the worksheet, enabling self-assessment and targeted revision. Cross-check your working carefully: even if your final answer is correct, review the method to ensure it matches standard board exam expectations, as partial marks depend on correct intermediate steps. For MCQs and fill-in-the-blanks, brief explanations clarify why one option is right and others wrong. For short and long answers, full working is shown with integral setup, evaluation, and simplification. Where applicable, alternative methods (such as using symmetry or substitution) are mentioned to broaden problem-solving strategies. Students preparing for the 2025 boards should revisit incorrect answers, redo them without looking at solutions, and verify understanding. CBSETUTOR.ai offers a 24×7 AI tutor where you can upload photos of your working and get instant, personalised feedback on errors—especially useful for diagnosing recurring mistakes in limit determination or sign handling, all at a flat ₹999/month for classes 6-12 with a 3-day free trial to get started risk-free.

Section A Answers: 1(b), 2(a), 3(a), 4(b), 5(c), 6(b)
Section B Answers: 7. πab, 8. 8a²/3, 9. 2, 10. (16π/3 - 8√2/3), 11. 1
Section C Answers: 12. False, 13. True, 14. True, 15. True
Section D & E: Full worked solutions below

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Tips for Scoring Full Marks in Application of Integrals

CBSE Class 12 Mathematics board examiners publish marking schemes that reveal common reasons for mark deduction in Chapter 8: missing or incorrect diagrams (costs 0.5-1 mark), wrong limits of integration (costs 1-2 marks), and arithmetic errors in final simplification (costs 0.5 mark). To maximise your score, always draw a neat, labelled sketch showing curves, axes, intersection points, and shaded region—even if the question does not explicitly ask for it; examiners award method marks for clear working. Next, write limits with reasoning: state 'curves intersect at x = a, x = b' and show the algebra. Use standard integral results from NCERT Tables to save time, but write them out once to show you know them. For area between curves, decide which lies above by testing a sample x-value in the interval. Split integrals at intersection points within the domain to avoid negative areas. Finally, always state final answers in square units and box them for visibility. Time management is critical: spend no more than 5 minutes on a 3-mark question and 8 minutes on a 4-mark question, leaving 10 minutes at the end for review. Practice 10-15 previous years' board questions from this chapter to internalise these habits, and use Class 12 Mathematics notes for quick formula revision the night before the exam.

Always draw and label a diagram, even if not asked—shows clear understanding
State intersection points explicitly: 'solving f(x)=g(x) gives x=...'
Use symmetry where applicable to halve computation (even functions, symmetric limits)
Double-check sign: area is always positive; use |f(x)-g(x)| or split at crossings
Write final answer in square units and box it for easy examiner identification
Allocate time strictly: 5 min for 3-mark, 8 min for 4-mark questions

Frequently asked questions

How many marks does Chapter 8 Application of Integrals carry in the CBSE Class 12 Mathematics board exam?+

Application of Integrals typically carries 5-6 marks in the CBSE Class 12 board exam, usually distributed as one 3-mark and one 4-mark question, or integrated into a case-study scenario worth 2 marks plus one short-answer question.

What are the most common mistakes students make in Application of Integrals?+

The three most frequent errors are: incorrect identification of integration limits when curves intersect, forgetting to take absolute value when a curve lies below the x-axis, and not sketching the region before setting up the integral, leading to wrong area formulas.

Is it compulsory to draw a diagram for area problems in board exams?+

While not always explicitly required, drawing a clear, labelled diagram is highly recommended. CBSE marking schemes often award 0.5 to 1 method mark for correct diagrams, and examiners view them as evidence of conceptual understanding, especially in 4-mark questions.

How do I decide whether to integrate with respect to x or y?+

Integrate with respect to x when the region is easily described by vertical strips (top curve minus bottom curve), and with respect to y when horizontal strips are simpler. Choose the variable that gives easier limits and simpler integrand expressions.

Can area between two curves ever be negative?+

No. Area is always a positive quantity. If your integral evaluates to a negative number, it means you have subtracted the curves in the wrong order. Use absolute value |f(x)-g(x)| or split the integral at points where curves cross to ensure positive results.

What are the key formulas I must memorise for this chapter?+

Essential formulas include: area under y=f(x) from x=a to b is ∫[a to b] f(x) dx; area between y=f(x) and y=g(x) is ∫[a to b] |f(x)-g(x)| dx; area of ellipse x²/a² + y²/b² = 1 is πab; and area of circle x²+y²=r² is πr². Know standard integrals of x^n, sin x, cos x, e^x, 1/x.

How can I find intersection points of two curves quickly?+

Set the two curve equations equal and solve for the variable. For y=f(x) and y=g(x), solve f(x)=g(x). For implicit curves, substitute one equation into the other. Always verify solutions by substituting back, and sketch to confirm intersection points lie in the region of interest.

Are NCERT exercise problems sufficient for board exam preparation in this chapter?+

NCERT exercises 8.1 and 8.2, plus miscellaneous exercise problems, form the backbone of board exam questions. However, supplement with NCERT Exemplar, previous years' board papers from 2020-2024, and sample papers released by CBSE to cover case-study formats and HOTS questions introduced recently.

What is the best strategy to manage time for Application of Integrals questions in the board exam?+

Allocate 5 minutes per 3-mark question and 8 minutes per 4-mark question. Spend the first minute sketching the region and identifying limits. Use standard integral results without deriving them. If stuck, move on and return later—do not let one area problem consume 15 minutes.

How does CBSETUTOR.ai help specifically with Application of Integrals doubts?+

CBSETUTOR.ai lets you upload a photo of any Application of Integrals problem—whether from this worksheet, NCERT, or your school test—and receive instant step-by-step solutions. The AI identifies if you made limit errors or sign mistakes, offers similar practice problems, and tracks your progress, all for ₹999/month with a 3-day free trial.

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