---
title: "Area using Green's theorem"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:area-using-greens-theorem-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/area-using-greens-theorem-three
site: BCT Engineering
---

# Area using Green's theorem

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2067 Magh
- Video: [Application of green's theorem to find area of ellipse](https://www.youtube.com/watch?v=WZuxaqx-9lM) — MathOdyssey
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/area-using-greens-theorem-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Use Green's theorem to find the area of the ellipse x²/a² + y²/b² = 1.

## Short answer

By Green's theorem with P=−y/2, Q=x/2, area = ½∮( x dy − y dx). For ellipse x=a cos t, y=b sin t, x dy − y dx = ab dt. Integrate 0 to 2π: area = ½·ab·2π = πab.

## Answer

**Green's theorem:** ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy.

Choose P = −y/2, Q = x/2. Then ∂Q/∂x − ∂P/∂y = 1/2 + 1/2 = 1, so

```text
Area A = ∬_R dx dy = ½ ∮_C (x dy − y dx)
```

For the ellipse x²/a² + y²/b² = 1 use x = a cos t, y = b sin t, 0 ≤ t ≤ 2π (anticlockwise):

```text
dx = −a sin t dt,   dy = b cos t dt
x dy − y dx = (a cos t)(b cos t) dt − (b sin t)(−a sin t) dt
            = ab (cos²t + sin²t) dt = ab dt

A = ½ ∫₀^2π ab dt = ½ · ab · 2π = πab
```

**Answer: Area of the ellipse = πab.**

## Check yourself

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

1. Which choice gives area via Green's theorem?
   A. P = y, Q = x
   B. P = x, Q = y
   C. P = -y/2, Q = x/2
   D. P = y/2, Q = x/2
   Answer: C. Then dQ/dx - dP/dy = 1. (Key point: Choice of P,Q)

2. Area equals:
   A. (1/2) contour of (x dy + y dx)
   B. contour of (x dx + y dy)
   C. contour of x dy + y dx
   D. (1/2) contour integral of (x dy - y dx)
   Answer: D. From Green's theorem with the above choice. (Key point: Area formula)

3. For the ellipse, x dy - y dx equals:
   A. ab dt
   B. ab sin t cos t dt
   C. a^2 b^2 dt
   D. (a+b) dt
   Answer: A. ab(cos^2 + sin^2) dt = ab dt. (Key point: Parametrization)

4. The area is:
   A. 2 pi a b
   B. pi a b
   C. pi (a+b)
   D. pi a^2 b^2
   Answer: B. (1/2) * ab * 2pi. (Key point: Result)

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