Class 12 · Chapter 8 · Calculus unit (35 of 80 marks)

Application of Integrals Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 47 board-style questions.

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47 questions
13 multiple choice, 3 assertion–reason, 10 very short answer, 12 short answer, 6 long answer, 3 case study
About 12 hours to master

Calculus unit: 35 of 80 theory marks (Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations).

NCERT textbook: Chapter 8 on ncert.nic.in (PDF)

Revision notes

Application of Integrals — revision notes

1. What is in the syllabus

Area under simple curves only: straight lines, circles, parabolas and ellipses in standard form (centre or vertex at the origin), together with regions cut off by straight lines. Area between two curves (e.g. two parabolas, circle and parabola) has been deleted.

2. Key results

Vertical strips: area between $y = f (x) \geq 0$ , the $x$ -axis and $x = a$ , $x = b$ is $\int_{a}^{b} y d x$ .
Horizontal strips: area between $x = g (y) \geq 0$ , the $y$ -axis and $y = c$ , $y = d$ is $\int_{c}^{d} x d y$ .
Region between a curve and a line: $\int (upper - lower) d x$ or $\int (right - left) d y$ , limits = points of intersection.
If the curve is below the axis, the integral is negative: area $= | \int_{a}^{b} y d x |$ . If the curve crosses the axis, split the interval at the crossing.
$\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a} + C$ .
Circle $x^{2} + y^{2} = a^{2}$ : area $π a^{2}$ . Ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ : area $π a b$ (from $4 \int_{0}^{a} \frac{b}{a} \sqrt{a^{2} - x^{2}} d x$ ).
Parabolic segment: the region between a parabola and a chord perpendicular to its axis is $\frac{2}{3}$ of the enclosing rectangle.

3. Method (write these steps in the exam)

Draw a rough graph and shade the region.
Find the intersection points (these are the limits).
Choose vertical or horizontal strips so that the region needs as few integrals as possible.
Use symmetry (double the first-quadrant part) where valid.
Integrate, then write the answer in sq units.

Worked example 1

Area bounded by $y^{2} = 4 x$ and $x = 1$ : by symmetry, $2 \int_{0}^{1} 2 \sqrt{x} d x = \frac{8}{3}$ sq units.

Worked example 2

Area between $y^{2} = 3 x$ and $y = x$ : they meet at $x = 0, 3$ . Area $= \int_{0}^{3} (\sqrt{3 x} - x) d x = 6 - \frac{9}{2} = \frac{3}{2}$ sq units.

Worked example 3

Area of the circle $x^{2} + y^{2} = 9$ between $x = 0$ and $x = \frac{3}{2}$ (upper half): ${[\frac{x}{2} \sqrt{9 - x^{2}} + \frac{9}{2} \sin^{- 1} \frac{x}{3}]}_{0}^{3 / 2} = \frac{9 \sqrt{3}}{8} + \frac{3 π}{4}$ .

Common errors

Reporting a negative or zero “area” because part of the curve is below the axis.
Forgetting to double when using symmetry — or doubling a region that is not symmetric.
Wrong limits: using the intercepts of the line instead of the intersection points with the curve.
Integrating $y$ with respect to $y$ (mixing up vertical and horizontal strips).
Dropping the $\frac{a^{2}}{2} \sin^{- 1} \frac{x}{a}$ term or evaluating $\sin^{- 1}$ in degrees.

Board-exam tips

A neat, labelled rough graph with the region shaded usually earns a mark in the 5-mark question.
Always write the final answer with “sq units” (or the actual units in a context question).
Check your answer against a simple bound (e.g. the enclosing rectangle or the whole circle).

Topics in this chapter: Area under a curve (vertical strips) · Areas of circles and ellipses · Area using horizontal strips · Region bounded by a curve and a line · Applications and modelling.

Formulas for this chapter: Chapter 8 on the Class 12 formula sheet (free PDF and A3 poster).

Application of Integrals Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 174 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceArea under lines and parabolas

The area of the region bounded by $y = x^{2}$ , the $x$ -axis and the lines $x = 0$ and $x = 3$ is

Show worked answer

$\begin{aligned} \int_{0}^{3} x^{2} d x & = {[\frac{x^{3}}{3}]}_{0}^{3} \\ = 9 \end{aligned}$ square units A1
Correct option: (a) $9$

Q2·1 mark·Multiple choiceArea of circles and ellipses

What is the area of the region inside the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1$ ?

Q3·1 mark·Assertion–reasonArea under lines and parabolas

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The area of the region bounded by $y = \sin x$ , the $x$ -axis and the lines $x = 0$ and $x = 2 π$ is $4$ square units.

Reason (R): $\int_{0}^{2 π} \sin x d x = 0$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerArea under lines and parabolas

Find the area of the region bounded by the line $y = 2 x + 1$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ .

Show worked answer

Answer:

10

square units

Area $= \int_{1}^{3} (2 x + 1) d x = [x^{2} + x]_{1}^{3}$ M1
$= 12 - 2 = 10$ square units A1

Q5·2 marks·Short answerArea under lines and parabolas

How much area lies under the curve $y = \sqrt{x}$ and above the $x$ -axis between $x = 0$ and $x = 9$ ?

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerArea of circles and ellipses

Use integration to find the area of the quarter of the circle $x^{2} + y^{2} = 36$ that lies in the first quadrant.

Show worked answer

Answer:

9 π

square units

Area $= \int_{0}^{6} \sqrt{36 - x^{2}} d x$ M1
$= {[\frac{x}{2} \sqrt{36 - x^{2}} + \frac{36}{2} \sin^{- 1} \frac{x}{6}]}_{0}^{6}$ A1
$= 18 \times \frac{π}{2} = 9 π$ square units A1

Q7·3 marks·Short answerArea under lines and parabolas

The part of the parabola $y^{2} = 9 x$ in the first quadrant, the $x$ -axis and the line $x = 4$ enclose a region. Find its area.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerArea of circles and ellipses

Using integration, find the area of the region enclosed by the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ .

Show worked answer

Answer:

20 π

square units

Upper half: $y = \frac{4}{5} \sqrt{25 - x^{2}}$ M1
By symmetry, area $= 4 \int_{0}^{5} \frac{4}{5} \sqrt{25 - x^{2}} d x$ M1
$\begin{aligned} \int_{0}^{5} \sqrt{25 - x^{2}} d x & = {[\frac{x}{2} \sqrt{25 - x^{2}} + \frac{25}{2} \sin^{- 1} \frac{x}{5}]}_{0}^{5} \\ = \frac{25 π}{4} \end{aligned}$ A1
Area $= 4 \times \frac{4}{5} \times \frac{25 π}{4}$ A1
$= 20 π$ square units A1

Q9·5 marks·Long answerArea with the y-axis

Find the area enclosed between the curve $x^{2} = 9 y$ (with $x \geq 0$ ), the $y$ -axis and the horizontal lines $y = 1$ and $y = 4$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyArea under lines and parabolas

Arched gateway. The opening of a garden gateway is bounded by the ground ( $y = 0$ ) and an arch with equation $y = 4 - \frac{x^{2}}{4}$ , with $x$ and $y$ in metres.

How wide is the opening at ground level? [1 mark]
How high is the top of the arch? [1 mark]
Find the area of the opening. [2 marks]

Show worked answer

$\begin{aligned} 4 - \frac{x^{2}}{4} = 0 \\ \Rightarrow & x = \pm 4 \end{aligned}$ : width $8$ m A1
Greatest $y$ is at $x = 0$ : $4$ m A1
Area $= \int_{- 4}^{4} (4 - \frac{x^{2}}{4}) d x = 2 {[4 x - \frac{x^{3}}{12}]}_{0}^{4}$ M1
$= 2 (16 - \frac{16}{3}) = \frac{64}{3} m^{2}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find the area under a curve with vertical strips and the areas of circles and ellipses using symmetry.

Read first: 1. What is in the syllabus; 2. Key results 15 practice questions · checkpoint: 4 questions, 8 marks, pass 80%

Practise step 1

Step 2

Board standard

You can choose vertical or horizontal strips and find the area between a curve and a line from their intersection points.

Read first: 2. Key results; 3. Method (write these steps in the exam); Worked examples 1-2 12 practice questions · checkpoint: 4 questions, 11 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write full-marks 5-mark area answers and case studies: sketch, shading, limits, integral and sq units.

Read first: 3. Method (write these steps in the exam); Board-exam tips 5 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle regions that need two integrals, parts of circles and ellipses and area questions where symmetry is only partial.

Read first: Worked example 3; Common errors 7 practice questions · checkpoint: 3 questions, 12 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 10 of the 47 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceArea under a curve (vertical strips)

The area of the region bounded by the parabola $y^{2} = 8 x$ and the line $x = 8$ is

(a) $\frac{256}{3}$ sq units
(b) $\frac{128}{3}$ sq units
(c) $\frac{64}{3}$ sq units
(d) $\frac{512}{3}$ sq units

Q2·1 mark·Multiple choiceArea under a curve (vertical strips)

If the area of the region bounded by the line $y = k x$ $(k > 0)$ , the $x$ -axis and the line $x = 3$ is $18$ sq units, then $k$ equals

(a) $2$
(b) $4$
(c) $6$
(d) $\frac{4}{3}$

Q3·2 marks·Very short answerArea under a curve (vertical strips)

Using integration, find the area of the region bounded by the line $x + 2 y = 8$ , the $x$ -axis and the lines $x = 2$ and $x = 6$ .

Worked solution·2 marks·Very short answerArea using horizontal strips

Using horizontal strips, find the area of the region ${(x, y) : x^{2} \leq 2 y, y \leq 4}$ .

Full worked solution Free, no sign-in

For $0 \leq y \leq 4$ the strip runs from $x = - \sqrt{2 y}$ to $x = \sqrt{2 y}$ : area $= \int_{0}^{4} 2 \sqrt{2 y} d y$ M1
$= 2 \sqrt{2} \cdot \frac{2}{3} \cdot 4^{3 / 2} = \frac{32 \sqrt{2}}{3}$ sq units A1

Every question here has a full worked solution. See them all →

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S128. All 24 →

Knowledge organiser: Application of Integrals on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Application of Integrals

No sketch, or an unshaded region, in the 5-mark question. Fix: draw a neat rough graph with the region shaded and intersection points labelled; it usually earns a mark.
Wrong limits: using a line's intercepts instead of its intersection points with the curve. Fix: solve the equations together and write the intersection points first.
Negative 'area' when the curve is below the axis. Fix: take the absolute value of each part separately, or integrate (upper − lower).
Doubling for symmetry when the region is not symmetric, or forgetting to double when it is. Fix: say which axis the region is symmetric about before multiplying by 2 or 4.
Dropping the (a²/2) sin⁻¹(x/a) term, or evaluating sin⁻¹ in degrees. Fix: write the full standard result and keep angles in radians.
No 'sq units' in the final answer. Fix: always finish with the units (or m² in a context question).

Exam strategy: how answers are step-marked · competency-based questions

Application of Integrals in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): question 33
Sample paper 2 (Board standard): question 29
Sample paper 3 (Above board): question 33
Sample paper 4 (Above board): question 33

Want it against the clock? Take a timed 30-mark chapter test on Application of Integrals, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on integrals, areas and differential equations.

Application of Integrals: common questions

What is in CBSE Class 12 Maths Chapter 8, Application of Integrals?

The topics are: Area under a curve (vertical strips), Areas of circles and ellipses, Area using horizontal strips, Region bounded by a curve and a line, Applications and modelling. This page has revision notes for each, and 47 board-style practice questions with step-marked solutions.

How many marks is Application of Integrals worth?

It is in the Calculus unit, which carries 35 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Application of Integrals?

No sketch, or an unshaded region, in the 5-mark question. Fix: draw a neat rough graph with the region shaded and intersection points labelled; it usually earns a mark. Wrong limits: using a line's intercepts instead of its intersection points with the curve. Fix: solve the equations together and write the intersection points first.

Are the Application of Integrals important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Application of Integrals — revision notes

1. What is in the syllabus

2. Key results

3. Method (write these steps in the exam)

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Application of Integrals Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Application of Integrals

Application of Integrals in our sample papers

Application of Integrals: common questions

Work through the Route to 95 with your progress saved