Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such round-about is made such that equation representing its boundary is given by C_1 : x² + y² = 64. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by C_2 : x² + y² = 4. Based on the given information, answer the following questions : (i) Represent the given equations C_1 and C_2 with the help of a diagram. (ii) Express y as a function of x, (y = f(x)), for both C_1 and C_2. (iii)(a) Using integration find the area of region covered by the roundabout. (iii)(b) Using integration, find the area of region covered by circular pond.

(ii) C_1: y = ±√(64-x²); C_2: y = ±√(4-x²); (iii)(a) 64π sq. units; (iii)(b) 4π sq. units

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2026: question 38

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedApplication of Integrals

Roundabouts are often made on busy roads to ease the traffic and avoid red lights.

One such round-about is made such that equation representing its boundary is given by $C_{1} : x^{2} + y^{2} = 64$ .

There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_{2} : x^{2} + y^{2} = 4$ .

Based on the given information, answer the following questions :

(i)
Represent the given equations $C_{1}$ and $C_{2}$ with the help of a diagram.
1 mark
(ii)
Express y as a function of $x$ , ( $y = f (x)$ ), for both $C_{1}$ and $C_{2}$ .
1 mark
(iii)(a)
Using integration find the area of region covered by the roundabout.
2 marks
OR
(iii)(b)
Using integration, find the area of region covered by circular pond.
2 marks

Figure in the paper: Aerial picture of a roundabout with a circular pond and fountain at its centre, surrounded by trees and roads. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Both circles have centre at the origin. Read the radius as the square root of the constant.

Hint 2

Solving $x^{2} + y^{2} = r^{2}$ for $y$ gives two branches, one for each sign.

Hint 3

Use the symmetry of the circle (4 × first-quadrant area) and the standard integral of $a^{2} - x^{2}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) Diagram: draw the $x$ - and $y$ -axes with origin O.
- Draw $C_{1}$ , a circle centred at O with radius 8, passing through $(\pm 8, 0)$ and $(0, \pm 8)$ .
- Draw $C_{2}$ , a smaller concentric circle with radius 2, passing through $(\pm 2, 0)$ and $(0, \pm 2)$ .
- Label both circles.
1 mark
(ii) For $C_{1}$ : $y = 64 - x^{2}$ (upper half) or $y = - 64 - x^{2}$ (lower half), with $- 8 \leq x \leq 8$ .
½ mark
(ii) For $C_{2}$ : $y = 4 - x^{2}$ or $y = - 4 - x^{2}$ , with $- 2 \leq x \leq 2$ .
½ mark
(iii)(a) The circle is symmetric about both axes, so the area is 4 times the area in the first quadrant:
Area $= 4 \int_{0}^{8} 64 - x^{2} d x$ .
1 mark
(iii)(a) Use $\int a^{2} - x^{2} d x = \frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$ :
$= 4 [\frac{x}{2} 64 - x^{2} + 32 sin^{- 1} \frac{x}{8}]_{0}^{8} = 4 [0 + 32 \cdot \frac{π}{2} - 0] = 64 π$ sq. units.
1 mark
(iii)(b) By symmetry, Area of pond $= 4 \int_{0}^{2} 4 - x^{2} d x$ .
1 mark
(iii)(b) $= 4 [\frac{x}{2} 4 - x^{2} + 2 sin^{- 1} \frac{x}{2}]_{0}^{2} = 4 [0 + 2 \cdot \frac{π}{2} - 0] = 4 π$ sq. units.
1 mark

Answer: (ii) $C_{1}$ : $y = \pm 64 - x^{2}$ ; $C_{2}$ : $y = \pm 4 - x^{2}$ ; (iii)(a) $64 π$ sq. units; (iii)(b) $4 π$ sq. units

Where the marks are

(i) Draws two concentric circles centred at O with radii 8 and 2, correctly labelled (1 mark)
(ii) Writes $y = \pm 64 - x^{2}$ for $C_{1}$ (½ mark)
(ii) Writes $y = \pm 4 - x^{2}$ for $C_{2}$ (½ mark)
(iii)(a) Sets up Area $= 4 \int_{0}^{8} 64 - x^{2} d x$ (1 mark)
(iii)(a) Evaluates to $64 π$ sq. units (1 mark)
(iii)(b) Sets up Area $= 4 \int_{0}^{2} 4 - x^{2} d x$ (1 mark)
(iii)(b) Evaluates to $4 π$ sq. units (1 mark)

Revise the chapter

Chapter: Application of Integrals (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Application of Integrals, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 38

Hints

Step-by-step solution

Revise the chapter

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