Application of Integrals Quick Lesson: Class 12 Maths Chapter 8 (NCERT)

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The chapter in one minute

What the chapter is about, and what the exam asks.

This chapter uses definite integrals to find areas: under a curve, inside a circle or an ellipse, and between a curve and a line.

The CBSE syllabus names lines, circles, parabolas and ellipses, in standard form only (like $x^2 + y^2 = a^2$ or $y^2 = 4ax$ ⁠). The book also uses a few other simple curves, like $\sin x$ ⁠.

Think of the region as many thin strips. A strip of height $y$ and width $dx$ has area $y\,dx$ ⁠, and the integral adds them all up.

The exam wants four things: a sketch with the region shaded, the limits (where the curve meets the axis, the given lines or the other graph), the area written as an integral, and the value in sq. units.

Most questions are one of seven kinds, and each kind has a fixed recipe. You will learn all seven below.

7question types

11formula cards

1sketch in every answer, with the region shaded

sq. unitsat the end of every area

Formulas

The formulas you must know

Learn these cards. Every question in the chapter is built from them.

Vertical strips

A = \int_a^b y\,dx

For a curve above the $x$ -axis, between the lines $x = a$ and $x = b$ ⁠. Take $y$ from the curve.

Horizontal strips

A = \int_c^d x\,dy

For a region between a curve and the $y$ -axis, on its right. Write the curve as $x$ in terms of $y$ ⁠, and use $y$ limits.

Below the axis

A = \left|\int_a^b y\,dx\right|

When the curve is below the axis from $a$ to $b$ ⁠, the integral comes out negative. An area is never negative, so take its size.

Crossing the axis

\begin{aligned} A &= \left|\int_a^c y\,dx\right| \\ &\quad + \left|\int_c^b y\,dx\right| \end{aligned}

The curve crosses the axis at $c$ ⁠: solve $y = 0$ ⁠. Take the size of each part, then add. The bars only change the part below the axis.

Powers of x

\begin{aligned} &\int_a^b x^n\,dx \\ &= \frac{b^{n + 1} - a^{n + 1}}{n + 1} \end{aligned}

For every power except −1. With a negative power, 0 must not lie between $a$ and $b$ ⁠.

One arch of sine

\int_0^{\pi}\sin x\,dx = 2

Worth remembering. Also $\int_0^{\pi/2}\cos x\,dx = 1$ ⁠.

The square-root card

\begin{aligned} &\int\sqrt{a^2 - x^2}\,dx \\ &= \frac{x}{2}\sqrt{a^2 - x^2} \\ &\quad + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C \end{aligned}

From Chapter 7. It does every circle and ellipse in this chapter.

Quarter circle

\begin{aligned} &\int_0^a\sqrt{a^2 - x^2}\,dx \\ &= \frac{\pi a^2}{4} \end{aligned}

At the top limit the root is 0, and $\sin^{-1}1 = \frac{\pi}{2}$ ⁠.

Circle of radius a

\begin{aligned} A &= 4\int_0^a\sqrt{a^2 - x^2}\,dx \\ &= \pi a^2 \end{aligned}

For $x^2 + y^2 = a^2$ ⁠. One quarter, times four.

Ellipse

\begin{aligned} A &= 4\int_0^a\frac{b}{a}\sqrt{a^2 - x^2}\,dx \\ &= \pi ab \end{aligned}

For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ⁠. The number under $x^2$ is $a^2$ ⁠, the one under $y^2$ is $b^2$ ⁠.

Curve and line

A = \int_a^b(\text{top} - \text{bottom})\,dx

Top and bottom are the $y$ of the upper and lower graph. $a$ and $b$ are where they meet.

Question types

The question types and how to crack each one

For each type: how to spot it, what to do, and one real NCERT question worked the exam way. Every answer starts with a sketch.

1. Area under a curve: vertical strips

Spot it: a curve $y = f(x)$ above the $x$ -axis, and two lines such as $x = 1$ and $x = 2$ ⁠.

Sketch the curve and the two lines, and shade the region down to the $x$ -axis.
Write the area as $\int_a^b y\,dx$ ⁠, with $y$ from the curve.
Integrate, put in the limits (top minus bottom), and write sq. units.

Miscellaneous Q1(i). Find the area under $y = x^2$ ⁠, $x = 1$ ⁠, $x = 2$ and the $x$ -axis.

Sketch: the parabola above the $x$ -axis, with the region from $x = 1$ to $x = 2$ shaded.

\begin{aligned} A &= \int_1^2 y\,dx = \int_1^2 x^2\,dx \\ &= \left[\frac{x^3}{3}\right]_1^2 = \frac{8}{3} - \frac{1}{3} \end{aligned}

∴ $A = \frac{7}{3}$ sq. units

See a full exam answer · Go deeper: see it visually · Basics: area as a sum of strips

In past papers (3): click to see

Board 2025 Q17 · Sample 2020-21 Q8 · Sample 2020-21 Q24

2. A region against the $y$ -axis: horizontal strips

Spot it: the region is bounded by the $y$ -axis, or the limits are $y$ values, like the line $y = 3$ ⁠.

Sketch. Mark where any given line meets the curve.
Write the curve as $x$ in terms of $y$ ⁠.
Find the area as $\int_c^d x\,dy$ ⁠, with the $y$ limits.

Exercise 8.1 Q4. Find the area bounded by $y^2 = 4x$ ⁠, the $y$ -axis and the line $y = 3$ ⁠.

Sketch: the parabola opens to the right. Shade between it and the $y$ -axis, from $y = 0$ up to $y = 3$ ⁠. The line $y = 3$ meets the curve at $x = \frac{9}{4}$ ⁠.

Use horizontal strips, with $x = \frac{y^2}{4}$ ⁠.

\begin{aligned} A &= \int_0^3 x\,dy = \int_0^3\frac{y^2}{4}\,dy \\ &= \left[\frac{y^3}{12}\right]_0^3 = \frac{27}{12} \end{aligned}

∴ $A = \frac{9}{4}$ sq. units, option (B)

See a full exam answer · Go deeper: see it visually · Basics: strips can be horizontal

In past papers (2): click to see

Sample 2025-26 Q23(b) · Sample 2024-25 Q18

3. Circles: the square-root card, one quarter at a time

Spot it: $x^2 + y^2 = a^2$ ⁠, a circle with its centre at the origin and radius $a$ ⁠.

Read off the radius $a$ ⁠. Sketch the circle and shade the part asked for.
Solve for $y$ ⁠. The top half is $y = \sqrt{a^2 - x^2}$ ⁠.
Integrate from 0 to $a$ with the square-root card. That is one quarter.
For the whole circle, multiply by 4.

Exercise 8.1 Q3. Find the area in the first quadrant bounded by $x^2 + y^2 = 4$ and the lines $x = 0$ and $x = 2$ ⁠.

Sketch: the quarter of the circle of radius 2 in the first quadrant, shaded. Its top edge is $y = \sqrt{4 - x^2}$ ⁠.

\begin{aligned} A &= \int_0^2\sqrt{4 - x^2}\,dx \\ &= \Big[\frac{x}{2}\sqrt{4 - x^2} \\ &\quad + 2\sin^{-1}\frac{x}{2}\Big]_0^2 \\ &= 2\sin^{-1}1 = 2\cdot\frac{\pi}{2} \end{aligned}

∴ $A = \pi$ sq. units, option (A)

See a full exam answer · Book Example 1 · Go deeper: see it visually · Basics: the quarter-circle integral

In past papers (4): click to see

Board 2026 Q38 · Sample 2026-27 Q34(a) · Sample 2021-22 T2 Q12(b) · Sample 2020-21 Q34(a)

4. Ellipses: read off $a$ and $b$ ⁠, then times four

Spot it: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ⁠, with numbers under $x^2$ and $y^2$ ⁠.

Read off $a$ and $b$ ⁠. Sketch the ellipse through $(\pm a, 0)$ and $(0, \pm b)$ ⁠, and shade the quarter in the first quadrant.
Solve for $y$ ⁠. The top edge is $y = \frac{b}{a}\sqrt{a^2 - x^2}$ ⁠.
Integrate from 0 to $a$ ⁠, then multiply by 4.
Check: the answer must be $\pi ab$ ⁠.

Exercise 8.1 Q1. Find the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ ⁠.

Here $a = 4$ and $b = 3$ ⁠. Sketch the ellipse through $(\pm 4, 0)$ and $(0, \pm 3)$ ⁠. Its top edge is $y = \frac{3}{4}\sqrt{16 - x^2}$ ⁠.

\begin{aligned} A &= 4\int_0^4\frac{3}{4}\sqrt{16 - x^2}\,dx \\ &= 3\Big[\frac{x}{2}\sqrt{16 - x^2} \\ &\quad + 8\sin^{-1}\frac{x}{4}\Big]_0^4 \\ &= 3\cdot 8\cdot\frac{\pi}{2} \end{aligned}

∴ $A = 12\pi$ sq. units

See a full exam answer · Book Example 2 · Go deeper: see it visually · Basics: one quarter, times four

In past papers (3): click to see

Board 2024 Q34 · Board 2022 T2 Q11(a) · Sample 2020-21 Q34(b)

5. A curve that dips below the axis: split where it crosses

Spot it: part of the curve is below the $x$ -axis between the limits, like $\sin x$ from 0 to $2\pi$ ⁠, or a line that crosses the axis.

Sketch. Find where the curve crosses the $x$ -axis: solve $y = 0$ ⁠.
Split the integral at each crossing.
Put bars round each part below the axis, so it counts as positive.
Add the parts.

Miscellaneous Q3. Find the area bounded by the curve $y = \sin x$ between $x = 0$ and $x = 2\pi$ ⁠.

Sketch: one arch above the axis from 0 to $\pi$ ⁠, and one below from $\pi$ to $2\pi$ ⁠. Shade both. The curve crosses the axis where $\sin x = 0$ ⁠, at $x = \pi$ ⁠.

\begin{aligned} A &= \int_0^{\pi}\sin x\,dx \\ &\quad + \left|\int_{\pi}^{2\pi}\sin x\,dx\right| \\ &= \Big[-\cos x\Big]_0^{\pi} \\ &\quad + \Big|\Big[-\cos x\Big]_{\pi}^{2\pi}\Big| \\ &= 2 + |-2| \end{aligned}

∴ $A = 4$ sq. units

See a full exam answer · Book Examples 3 and 4 · Go deeper: see it visually · Basics: area below the axis

In past papers (2): click to see

Board 2022 T2 Q11(b) · Sample 2024-25 Q32

6. A modulus inside the integral: split at the corner

Spot it: bars inside the integral, like $|x + 3|$ or $x|x|$ ⁠.

Find where the inside of the bars is 0. That is the corner of the graph.
On each side, write the function without bars.
Split the integral at the corner, and add the two parts.
If asked what the value means: it is the area between the graph and the $x$ -axis, between the two limits.

Miscellaneous Q2. Sketch the graph of $y = |x + 3|$ and evaluate $I = \int_{-6}^{0}|x + 3|\,dx$ ⁠.

Sketch: a V with its corner at $(-3, 0)$ ⁠, where $x + 3 = 0$ ⁠. Shade from $x = -6$ to $x = 0$ ⁠.

For $x \lt -3$ ⁠, $|x + 3| = -(x + 3)$ ⁠. For $x \ge -3$ ⁠, $|x + 3| = x + 3$ ⁠.

\begin{aligned} I &= -\int_{-6}^{-3}(x + 3)\,dx \\ &\quad + \int_{-3}^{0}(x + 3)\,dx \\ &= -\Big[\frac{(x + 3)^2}{2}\Big]_{-6}^{-3} \\ &\quad + \Big[\frac{(x + 3)^2}{2}\Big]_{-3}^{0} \\ &= \frac{9}{2} + \frac{9}{2} \end{aligned}

∴ $I = 9$

See a full exam answer · Go deeper: see it visually · Basics: area below the axis

In past papers (6): click to see

Board 2026 Q10 · Board 2025 Q31 · Board 2024 Q28(b) · Sample 2025-26 Q28(a) · Sample 2022-23 Q28(b) · Sample 2021-22 T2 Q11

7. A curve and a line: top minus bottom

Spot it: the region is shut in by a curve and a line, like a parabola and $y = x$ ⁠, sometimes written as a set with $\le$ signs. Your book has no worked question of this kind, but CBSE sample papers ask it.

Sketch both graphs. Solve the two equations together to find where they meet. These are your limits.
Between the meeting points, see which graph is on top.
Find the area as $\int_a^b(\text{top} - \text{bottom})\,dx$ ⁠.

CBSE sample paper 2025-26, Q28(B). Using integration, find the area of the region $\{(x, y) : x^2 - 4y \le 0,\ y - x \le 0\}$ ⁠.

Read the set: $x^2 - 4y \le 0$ means $y \ge \frac{x^2}{4}$ ⁠, on or above the parabola $x^2 = 4y$ ⁠. And $y - x \le 0$ means $y \le x$ ⁠, on or below the line $y = x$ ⁠.

Sketch: the parabola opens upwards from $(0, 0)$ ⁠, and the line goes through the origin. Shade the region between them. They meet where $\frac{x^2}{4} = x$ ⁠, so at $(0, 0)$ and $(4, 4)$ ⁠.

From 0 to 4 the line is on top.

\begin{aligned} A &= \int_0^4\Big(x - \frac{x^2}{4}\Big)\,dx \\ &= \Big[\frac{x^2}{2} - \frac{x^3}{12}\Big]_0^4 \\ &= 8 - \frac{16}{3} \end{aligned}

∴ $A = \frac{8}{3}$ sq. units

Go deeper: area as a sum of strips · Basics: solve for $y$ to get each edge

In past papers (10): click to see

Board 2023 Q35 · Board 2022 T2 Q11(a) · Sample 2026-27 Q34(a) · Sample 2026-27 Q34(b) · Sample 2025-26 Q28(b) · Sample 2023-24 Q32 · Sample 2022-23 Q32 · Sample 2021-22 T2 Q12(a) · Sample 2021-22 T2 Q12(b) · Sample 2020-21 Q34(a)

Shortcuts

Shortcuts and quick tips

Small habits that save time and marks.

Your sketch must show: both axes, the curve, the given lines, the region shaded, and every meeting point with its coordinates. It need not be to scale, but CBSE asks for neat, clean figures.
Sketch first, always. It shows you the limits and whether any part is below the axis.
To find where a curve meets the $x$ -axis, put $y = 0$ ⁠. To find where it meets a line, solve the two equations together. For a line such as $y = 3$ ⁠, just put 3 into the curve.
Circle or ellipse: work out one quarter, from 0 to $a$ ⁠, and multiply by 4.
Check your answer with $\pi a^2$ for a circle and $\pi ab$ for an ellipse. In the exam, still show the integral.
At the top limit of the square-root card, the root part is 0 and $\sin^{-1}1 = \frac{\pi}{2}$ ⁠. Only one term is left.
In an MCQ, the wrong options are often the plain integral, where the part below the axis cancels. An area is never negative.
Keep $\pi$ as $\pi$ ⁠. Write $12\pi$ sq. units, not a decimal.

Watch out

Mistakes that cost marks

Each one is common, and each has an easy fix.

No sketch → always draw one; the figure is a mark on its own.
Integrating $\sin x$ from 0 to $2\pi$ in one go and getting 0 → split at $\pi$ ⁠; the area is 4.
Bottom minus top for a curve and a line → take top minus bottom. A negative answer means you swapped them.
Forgetting the 4 for a whole circle or ellipse → the integral from 0 to $a$ is only one quarter.
Mixing up $a$ and $b$ in an ellipse → $a^2$ is under $x^2$ ⁠. For $\frac{x^2}{4} + \frac{y^2}{9} = 1$ ⁠, $a = 2$ and $b = 3$ ⁠.
Using $dx$ for a region against the $y$ -axis → write $x$ in terms of $y$ and use $dy$ with $y$ limits.
Taking the minus root above the axis → there $y$ is positive: $y = +\sqrt{a^2 - x^2}$ ⁠.
No units → end with sq. units.

Marks

How the examiner gives marks

From CBSE's own marking schemes, here for a 5-mark area question. Write these lines and the marks follow.

The figure: the curve, the given lines and the shaded region. CBSE's model answer gives 1 mark for a correct figure.
The points where the curve meets the axis, the given lines or the other graph, with coordinates: 1 mark.
The area written as an integral (or a sum of integrals, with bars on any part below the axis): 1 mark.
The antiderivative in square brackets with its limits: 1 mark.
The value, ending in sq. units: 1 mark.
A slip is penalised only once, so keep going. An MCQ earns 1 mark for the right option only.

Practice

Practise these first

Easiest first. Together they cover every type above.

Miscellaneous Q1(ii) area under $y = x^4$ with vertical strips
Exercise 8.1 Q2 an ellipse: read off $a$ and $b$
Example 1 a whole circle: one quarter, times four
Example 2 an ellipse, also done with horizontal strips
Example 3 a line that crosses the axis: split there
Example 4 $y = \cos x$ ⁠: three parts, one below the axis
Miscellaneous Q5 $y = x|x|$ ⁠: a modulus, split at 0
Miscellaneous Q4 $y = x^3$ ⁠: spot the trap option
CBSE sample paper 2026-27, Q34(B) the curve $y^2 = 2x$ and the line $x - 2y = 0$ ⁠: top minus bottom. It is not in your book, so here is the answer to check: $\frac{16}{3}$ sq. units.

Seven kinds of question, one recipe each. Sketch first, find the limits, integrate, and write sq. units. Work through the practice list and you are ready for the exam.

Ready to go deeper? Open the interactive lesson

Areas, the quick way

The chapter in one minute

The formulas you must know

The question types and how to crack each one

1. Area under a curve: vertical strips

2. A region against the $y$ -axis: horizontal strips

3. Circles: the square-root card, one quarter at a time

4. Ellipses: read off $a$ and $b$ ⁠, then times four

5. A curve that dips below the axis: split where it crosses

6. A modulus inside the integral: split at the corner

7. A curve and a line: top minus bottom

Shortcuts and quick tips

Mistakes that cost marks

How the examiner gives marks

Practise these first

You can do this

Areas, the quick way

The chapter in one minute

The formulas you must know

The question types and how to crack each one

1. Area under a curve: vertical strips

2. A region against the yyy-axis: horizontal strips

3. Circles: the square-root card, one quarter at a time

4. Ellipses: read off aaa and bbb⁠, then times four

5. A curve that dips below the axis: split where it crosses

6. A modulus inside the integral: split at the corner

7. A curve and a line: top minus bottom

Shortcuts and quick tips

Mistakes that cost marks

How the examiner gives marks

Practise these first

You can do this

2. A region against the $y$ -axis: horizontal strips

4. Ellipses: read off $a$ and $b$ ⁠, then times four