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

# Area using Green's theorem

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

- Asked in: 2082 Baishakh, 5 marks; 2078 Bhadra, 5 marks; 2069 Chaitra, 5 marks; 2069 Asar, 5 marks (asked 4 times)
- Video: [Finding Areas with Green’s Theorem](https://www.youtube.com/watch?v=_aEgUhb6LtA) — Mathemation
- 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Use Green's theorem to determine the area of the curve (x/a)^(2/3) + (y/b)^(2/3) = 1.

## Short answer

Parametrize by x = a cos³t, y = b sin³t. Using area = ½ ∮ (x dy − y dx) and simplifying gives x dy − y dx = 3ab sin²t cos²t dt, so area = 3πab/8 square units.

## Answer

**Green's theorem:** ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy for a simple closed curve C traversed counter-clockwise.

#### Area formula
Choosing P = −y/2, Q = x/2 gives ∂Q/∂x − ∂P/∂y = 1, so

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

#### The curve (x/a)^(2/3) + (y/b)^(2/3) = 1 (a stretched astroid)
Parametrize: x = a cos³t, y = b sin³t, 0 ≤ t ≤ 2π. This satisfies cos²t + sin²t = 1.

```text
dx = −3a cos²t sin t dt
dy =  3b sin²t cos t dt

x dy − y dx = a cos³t · 3b sin²t cos t dt + b sin³t · 3a cos²t sin t dt
            = 3ab sin²t cos²t (cos²t + sin²t) dt
            = 3ab sin²t cos²t dt = (3ab/4) sin²2t dt
```

```text
Area = ½ ∫₀^{2π} (3ab/4) sin²2t dt
     = (3ab/8) ∫₀^{2π} (1 − cos 4t)/2 dt
     = (3ab/8) · π
```

**Answer: Area = 3πab/8 square units.** (For a = b this reduces to the astroid area 3πa²/8.)

## Check yourself

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

1. Curve (x/a)^(2/3)+(y/b)^(2/3)=1 is:
   A. x=a cos t, y=b sin t
   B. x=a cos³t, y=b sin³t
   C. x=a sin³t, y=a cos³t
   D. x=b cos³t, y=a sin³t
   Answer: B. Stretched astroid. (Key point: Parametrisation)

2. Area =
   A. ½∮(x dy − y dx)
   B. ∮ x dx
   C. ½∮(x dx + y dy)
   D. ∮ y dy
   Answer: A. From P=−y/2, Q=x/2. (Key point: Area formula)

3. x dy − y dx =
   A. ab dt
   B. 3ab cos⁴t dt
   C. ab sin t dt
   D. 3ab sin²t cos²t dt
   Answer: D. Using cos²+sin²=1. (Key point: Integrand)

4. Area:
   A. πab
   B. 3πab/4
   C. 3πab/8
   D. πab/8
   Answer: C. Reduces to 3πa²/8 for a=b. (Key point: Result)

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