---
title: "Smaller area bounded by circle and line"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:smaller-area-bounded-circle-line"
source: https://www.bctengineer.com/q/semester_2/mathematics/smaller-area-bounded-circle-line
site: BCT Engineering
---

# Smaller area bounded by circle and line

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2071 Magh, 5 marks
- Video: [Find the area of the smaller part of the circle x² + y² = 9 cut off by the line x + y = 3](https://www.youtube.com/watch?v=-4h6OIuMG9c) — Mathematics Untold
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/smaller-area-bounded-circle-line
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Find by double integration the smaller of the areas bounded by the circle x²+y² = 9 and the line x+y = 3.

## Short answer

The smaller area is the segment cut off by the chord x+y=3 from the circle x²+y²=9. Integrating dy dx over 0≤x≤3 from y=3-x to y=√(9-x²) gives area = 9(π-2)/4 square units.

## Answer

**Curves.** The circle x² + y² = 9 (centre origin, radius 3) and the line x + y = 3.

#### Intersection
Put y = 3 − x: x² + (3 − x)² = 9 → 2x² − 6x = 0 → x = 0 or x = 3. Points: (0, 3) and (3, 0).

The line cuts off a small segment from the circle (the part between the chord and the arc in the first quadrant). The line is at distance 3/√2 ≈ 2.12 from the centre, less than 3, so a segment is cut; the smaller area is the one not containing the centre.

```text
y
3 *
  | '.
  |   '.  segment
  |     '.
  |_______*___ x
 O        3
 (chord x + y = 3, arc of circle)
```

#### Area
For 0 ≤ x ≤ 3, y runs from the line (y = 3 − x) up to the arc (y = √(9 − x²)).

```text
A = ∫₀³ ∫_{3−x}^{√(9−x²)} dy dx
  = ∫₀³ ( √(9 − x²) − 3 + x ) dx
```

Now ∫₀³ √(9 − x²) dx = quarter circle area = (1/4)π(3)² = 9π/4, and ∫₀³ (x − 3) dx = 9/2 − 9 = −9/2.

```text
A = 9π/4 − 9/2 = 9(π − 2)/4 ≈ 2.57
```

**Answer: smaller area = 9(π − 2)/4 square units.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Circle x²+y²=9 and line x+y=3 meet at:
   A. (0,3),(3,0)
   B. (1,2),(2,1)
   C. (3,3)
   D. (0,0),(3,3)
   Answer: A. 2x²−6x=0. (Key point: Intersection)

2. Smaller area integrand:
   A. √(9−x²)
   B. 3−x
   C. (3−x) − √(9−x²)
   D. √(9−x²) − (3−x)
   Answer: D. Arc above chord. (Key point: Setup)

3. Area:
   A. 9(π−2)/4
   B. 9π/4
   C. 9(π+2)/4
   D. 9/2
   Answer: A. Quarter circle minus triangle. (Key point: Result)

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