Application of Integrals Class 12 Notes - Class 12 Maths Ch 8

The the resource Notes provided here cover every section of Class 12 Mathematics Chapter 8 Application of Integrals. The chapter notes Notes retain the formal structure of theorem and proof, but supplement each result with the practical method used to apply it in CBSE numerical questions.

• CBSE Class 12 Boards: 5 to 7 marks on its own, almost always in the form of one 5-mark long-answer area question; combined with Integrals the calculus unit returns 9 to 12 marks every year.
• JEE Main: 1 to 2 questions every session (around 3-4% of the Maths section), usually a region bounded by two curves or a circle-and-parabola pairing.
• CUET UG Maths/Applied Maths: Expect 2 to 4 MCQs from area-under-curve and area-between-two-curves combined, frequently testing limit setup and symmetry shortcuts.

The the PDF Notes below cover every Application of Integrals sub-topic the CBSE Class 12 Maths paper draws from, set-up of the integral, sketching the bounded region, choosing the right limits, and the symmetry trick that halves your working time. You can cross-check the solved steps against the Chapter 8 NCERT Solutions.

These this chapter Notes have been built by Class 12 Maths specialists, mapped to the latest NCERT 2026-27 edition, and refined against the last five years of CBSE board and JEE Main papers so what you revise is what the exam tends to ask.

What you will learn from these Class 12 the Application of Integrals Class 12 Notes

We have structured this set of notes around three goals, speed of set-up, geometric intuition, and accuracy in the final evaluation. Most students lose marks not in the integration step but in the sketch, the limits, or the sign of the area.

• NCERT-mapped flow: Every sub-topic follows the order of sections 8.1 and 8.2 in the Class 12 Maths textbook so revising alongside the book takes seconds.
• Sketch-first workflow: Every formula is paired with the rough graph you should draw, because a wrong sketch is the single biggest cause of zero marks on a 5-mark area question.
• Boards plus JEE in one place: Standard board curves (parabola, circle, ellipse) sit alongside the JEE flavour (mod functions, intersecting parabolas, parametric regions).
• Mistake-proofing built in: Sign of the integrand below the x-axis, the modulus trick, and the swap-to-dy decision rule are all flagged inline so you stop repeating the errors examiners see every year.

Application of Integrals Video Walkthrough

Source: Magnet Brains on YouTube

Class 12 Maths Chapter 8 Application of Integrals: Topic-by-Topic Breakdown

The chapter is short on theory and heavy on diagrams. Below is every concept broken down with the formula, the standard board use case, and a quick tip for avoiding the most common slip.

Area under a Simple Curve (Area w.r.t. the x-axis)

If y = f(x) is a continuous, non-negative function on [a, b] , then the area of the region bounded by the curve y = f(x) , the x-axis, and the ordinates x = a and x = b is

$$ A = \int_{a}^{b} y\, dx = \int_{a}^{b} f(x)\, dx $$

This is the workhorse formula of these notes Notes. Think of it as the limit of a Riemann sum: divide [a, b] into thin vertical strips of width dx , each strip has height y = f(x) , so each strip's area is y dx , and the integral adds them all up.

Memory Hook: The strip is vertical, the variable of integration is x, and the limits are the x-coordinates where the region starts and ends. Three things lined up, never break the pattern.

Area under a Curve w.r.t. the y-axis (Horizontal Strips)

When the curve is given as x = g(y) , or when integrating with respect to x would split the region into two awkward parts, you swap to horizontal strips. The area between x = g(y) , the y-axis, and the abscissae y = c and y = d is

$$ A = \int_{c}^{d} x\, dy = \int_{c}^{d} g(y)\, dy $$

Each strip is now horizontal, width dy , and length x = g(y) . This is the formula CBSE tests when the boundary is the y-axis (for example, the region bounded by y² = 4x , the y-axis, and y = 1, y = 3 ).

Decision Rule: If the curve is easier to write as x = g(y) than as y = f(x) , or if a single vertical strip would cross more than one curve, use dy. Otherwise stick with dx.

Region Lying Below the x-axis (Sign of the Integral)

If part of the curve dips below the x-axis on [a, b] , the definite integral _a^b f(x) dx returns a negative value for that portion. Area, however, is always positive. So when the curve crosses the x-axis at some point c ∈ (a, b) :

$$ A = \left| \int_{a}^{c} f(x)\, dx \right| + \left| \int_{c}^{b} f(x)\, dx \right| $$

Split the integral at every zero of f(x) , evaluate each piece, take the absolute value, and add. Skipping the modulus is the most common reason a perfectly correct integration loses 2 marks.

Area between Two Curves (the 5-Mark Board Question)

This is the highest-value sub-topic in the this Class 12 page Notes. If two curves y = f(x) (upper) and y = g(x) (lower) intersect at x = a and x = b , the area enclosed between them is

$$ A = \int_{a}^{b} \left[ f(x) - g(x) \right] dx $$

The procedure is mechanical and worth memorising:

1. Sketch both curves on the same axes, even a rough sketch is enough.
2. Find the points of intersection by solving f(x) = g(x) . These give the limits a and b .
3. Identify the upper curve on the bounded region (the one with the larger y-value at any sample x inside the interval).
4. Integrate (upper - lower) between the intersection points.

If the curves are easier to express as functions of y, swap to A = _c^d [x_right(y) - x_left(y)] dy instead.

Board Tip: Always write the intersection step explicitly, even if the points are obvious from the sketch. Examiners award 1 mark just for the correct limits.

Symmetry Shortcut for Standard Curves

Most CBSE-style regions (circles, ellipses, parabolas with axis along x or y) are symmetric about one of the axes. Use this aggressively:

• Circle x² + y² = a² , symmetric about both axes, area of a quarter equals 14 π a² , so total area is computed from 4 ₀^a √a² - x² dx .
• Ellipse x²a² + y²b² = 1 , symmetric about both axes, total area = 4 ₀^a ba√a² - x² dx = π a b .
• Parabola y² = 4ax , symmetric about the x-axis, so a region bounded above and below by the parabola halves the work.

Quick Note: Doubling or quadrupling at the end is risk-free once the symmetry is established. Forgetting to multiply back is a 1-mark slip examiners actively look for.

Standard Region Templates Worth Memorising

Five specific regions reappear across CBSE, JEE Main and CUET. Memorise each as a finished result so you can spot the question type within seconds of reading it.

• Quarter-circle bounded by x² + y² = a² , x-axis, y-axis in the first quadrant: area = π a²4 .
• Region under y = x² between x = 0 and x = a : area = a³3 .
• Region between y² = 4ax and y = 2ax (parabola and chord through origin): area = 4a²3 (a JEE favourite).
• Region between y = x² and y = x in [0, 1] : area = 16 .
• Region inside circle x² + y² = 4 and above y = x : area = π - 2 (recurring CBSE board variant).

solved examples: Area Bounded by a Parabola and a Line

Find the area of the region bounded by y² = 4x and the line x = 4 .

Step 1, sketch: The parabola opens rightward with vertex at the origin; the line x = 4 is vertical, cutting the parabola at (4, 4) and (4, -4) .

Step 2, by symmetry about the x-axis, total area = 2 ₀⁴ y dx = 2 ₀⁴ 2√x dx .

Step 3, evaluate: 2 · 2 ₀⁴ x^1/2 dx = 4 · [ 23 x^3/2 ]₀⁴ = 4 · 23 · 8 = 643 square units.

solved examples: Area Between Two Curves Using dy

Find the area enclosed by y² = x and x = 2 - y² .

Step 1, intersections: Set y² = 2 - y² ⇒ y² = 1 ⇒ y = ± 1 . The curves intersect at (1, 1) and (1, -1) .

Step 2, both curves are easier as functions of y, so use horizontal strips. The right boundary is x = 2 - y² , the left boundary is x = y² .

Step 3, set up: A = _-1¹ [(2 - y²) - y²] dy = _-1¹ (2 - 2y²) dy .

Step 4, evaluate using symmetry: 2 ₀¹ (2 - 2y²) dy = 2 [ 2y - 2y³3 ]₀¹ = 2 ( 2 - 23 ) = 83 square units.

NCERT Class 12 Maths Chapter 8 Application of Integrals: Important Topics

The table below distils the chapter to the eight sub-topics CBSE and competitive exams keep returning to. Tick each off as you finish revising.

Application of Integrals: Most Repeated Questions in CBSE Board Exams

A 5-year scan of CBSE Class 12 Maths papers shows the same four question shapes reappearing. Practise each at least twice before your board exam.

Ques. Find the area of the region bounded by the curve y² = x and the line y = x . (2019, 2023)

[5-Mark Question]

Intersections: y = x ⇒ x² = x ⇒ x = 0, 1 . Sketch shows the parabola y = √x lies above the line y = x on [0, 1] . So A = ₀¹ (√x - x) dx = [ 23x^3/2 - x²2 ]₀¹ = 23 - 12 = 16 square units.

Ques. Find the area of the region bounded by the ellipse x²16 + y²9 = 1 . (2020, 2024)

[5-Mark Question]

Here a = 4, b = 3 . Using symmetry, total area = 4 ₀⁴ y dx = 4 ₀⁴ 34√16 - x² dx = 3 ₀⁴ √16 - x² dx .

Using the standard result ₀^a √a² - x² dx = π a²4 , the area equals 3 · π · 164 = 12π square units. (Or directly: π a b = 12π .)

Ques. Find the area of the region bounded by y = x² and y = |x| . (2018, 2022)

[5-Mark Question]

Using symmetry about the y-axis, compute on [0, 1] and double. Intersections in [0, 1] : x² = x ⇒ x = 0, 1 .

On this interval y = x lies above y = x² , so A = 2 ₀¹ (x - x²) dx = 2 [ x²2 - x³3 ]₀¹ = 2 · 16 = 13 square units.

Ques. 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 . (2017, 2023)

[5-Mark Question]

The line y = x meets the circle at (4, 4) . Split the region at x = 4 . Area = ₀⁴ x dx + ₄^√32 √32 - x² dx = 8 + [ x2√32 - x² + 16 sin^-1x√32 ]₄^√32 = 4π square units.

Application of Integrals in Recent CBSE Papers, A 5-Year Analysis

Across the last five board cycles, the the resource Notes has shown a stable pattern, exactly one long-answer area question every paper, almost always set on a region bounded by a standard curve (parabola, circle, ellipse) and a simple line. The MCQs that began appearing from 2022 onward draw mainly from area-under-a-curve set-up.

• A 5-mark area-between-two-curves question has appeared in every one of the last 6 CBSE papers (2019, 2020, 2022, 2023, 2024, 2025).
• Parabola-and-line pairings appear most often (2019, 2022, 2025), followed by circle-and-line (2017, 2023) and ellipse (2020, 2024).
• Pure area-under-a-single-curve questions are now rare, the trend is firmly toward two-curve regions.
• Symmetry-based shortcuts save 2 to 3 minutes per question, particularly on ellipse and circle integrals.

Source:

• CBSE Class 12 Maths Set 1 Question Paper 2025
• CBSE Class 12 Maths Set 2 Question Paper 2025
• CBSE Class 12 Maths Set 3 Question Paper 2024

Weightage of Application of Integrals in JEE Main and CUET

Beyond boards, Application of Integrals continues to be a high-yield chapter in JEE Main, where every session paper carries at least one area question, almost always set on a pair of curves rather than a single curve. JEE Main doesn't currently include Maths, so the relevant competitive exams for the chapter notes Notes are JEE Main, JEE Advanced and CUET UG Maths.

• In JEE Main 2025 and 2024, every session paper carried at least one area-between-curves problem; the favoured pairings were parabola-line and parabola-parabola.
• CUET UG Maths usually includes 2 to 4 questions from this chapter, mostly set-up MCQs and limit-identification MCQs.
• The JEE flavour leans on regions that require splitting the limits or swapping to dy, the board flavour stays closer to symmetric, single-formula regions.

How will Collegedunia's Class 12 Maths the Application of Integrals Class 12 Notes help you?

These the PDF Notes are designed around the way the Class 12 Maths paper actually tests this chapter, one inevitable 5-mark area question, set on a region that always involves either a parabola, a circle, or an ellipse paired with a simple line or another curve.

• Sketch-first walk-throughs: Every solved examples starts from the rough graph, because the limits and the upper/lower decision both come from there.
• Both integration variables side by side: The dx and dy formulas are presented as a pair, with a decision rule for picking the right one, not as two separate sections you have to remember to compare.
• Standard-region cheat-sheet: Five reusable region templates (quarter-circle, parabola-chord, parabola-parabola, circle-line, ellipse) are pre-solved, so you arrive at the exam with the answer-shape already in your head.
• Board-paper alignment: Each solved question is tagged with the years CBSE has set it, so revision time is spent on what actually recurs, not on edge cases.

Application of Integrals Formulas at a Glance

The short list below is the entire formula stack you need for the chapter. Memorise the four core forms; everything else is symmetry and substitution.

• Area w.r.t. x-axis: A = _a^b y dx = _a^b f(x) dx
• Area w.r.t. y-axis: A = _c^d x dy = _c^d g(y) dy
• Area between two curves (vertical strips): A = _a^b [f(x) - g(x)] dx , where f is the upper curve.
• Area between two curves (horizontal strips): A = _c^d [x_R(y) - x_L(y)] dy , where x_R is the right curve.
• Region crossing x-axis: A = ∑ | _xi^x_i+1 f(x) dx | , splitting at every zero of f.
• Standard circle area: ₀^a √a² - x² dx = π a²4 .
• Standard ellipse area: π a b (full ellipse x²a² + y²b² = 1 ).

Full learn sheet: Class 12 Maths Chapter 8 Application of Integrals Formula Sheet

Common Mistakes Students Make When Using the Application of Integrals Class 12 Notes

Most errors in this chapter are not from lack of knowledge, they are from a handful of specific blind spots. Address these and your accuracy on the 5-mark question jumps sharply.

• Skipping the sketch: Students set up ∫ (upper - lower) without drawing the region and end up subtracting the wrong way, losing the entire 5-mark question to a sign error.
• Forgetting the modulus when the curve dips below the x-axis: _-1² (x³ - x) dx is not the area, you must split at x = 0, ± 1 and take absolute values.
• Confusing limits with intersection x-values vs y-values: A region bounded by y² = 4x and y = x is solved differently when integrated as dx (limits 0 to 4) versus dy (limits 0 to 4 also, but with swapped boundaries) , students routinely mix the two and lose 2 marks.
• Forgetting the symmetry factor: Computing area on the first quadrant for an ellipse and then forgetting to multiply by 4 at the end is the most common 1-mark slip in this chapter.
• Wrong upper curve on the bounded region: Two curves often swap which is "above" at different intervals. Always plug in a sample x -value inside the bounded interval, not at the intersection points, to decide.
• Leaving the answer without units: Area must be in square units. A bare "= 8" loses 0.5 marks on a 5-mark question because the examiner can't tell whether you understood it was an area.

Related Resources for Class 12 Maths Chapter 8

• NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals
• Class 12 Maths Chapter 8 Application of Integrals Formula Sheet
• Class 12 Maths Chapter 8 Application of Integrals Handwritten Notes
• NCERT Class 12 Maths Chapter 8 Application of Integrals Book PDF
• NCERT Exemplar Solutions for Class 12 Maths Chapter 8 Application of Integrals

NCERT Notes for Class 12 Maths: All Chapters

A quick jump-table to the rest of the Class 12 Maths Notes series, useful for unit-wise revision in the final week before the board exam.

this chapter Notes: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.

Application of Integrals Class 12 Notes - Quick Summary

• These notes Notes cover every section of Class 12 Mathematics Chapter 8 Application of Integrals, aligned to the 2026-27 NCERT print.
• The this Class 12 page Notes include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
• The the resource Notes are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.

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.

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:

The this chapter 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 these notes 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:

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.

• this Class 12 page 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 the resource 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:

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

The chapter 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:

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 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 these notes all formulas - it lists every identity used in the chapter.
• State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page 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:

How to Use the Application of Integrals Notes Page Most Effectively

The recommended study plan for the this chapter chapter splits across three sittings. The table below outlines what to do in each:

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.