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

# Apply Green's theorem

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

- Asked in: 2071 Shrawan, 5 marks
- Video: [Green's Theorem | Numerical | line Integral | Vector Calculus | Maths | in हिन्दी](https://www.youtube.com/watch?v=8hwyPfjosL4) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/apply-greens-theorem-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Using Green's theorem, evaluate ∮_C (y³ dx − x³ dy) where C is the boundary of the circle x² + y² = 4.

## Short answer

∂Q∂x − ∂P∂y = −3(x²+y²). In polar coordinates: ∮ = −3 ∫₀²π dθ ∫₀² r³ dr = −3·2π·4 = −24π.

## Answer

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

```text
P = y³,  Q = −x³
∂Q/∂x = −3x²,  ∂P/∂y = 3y²
∂Q/∂x − ∂P/∂y = −3(x² + y²)
```

R is the disc x² + y² ≤ 4. Use polar coordinates: x² + y² = r², dx dy = r dr dθ, 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π.

```text
∮_C = −3 ∫₀^2π ∫₀² r² · r dr dθ
    = −3 · 2π · [r⁴/4]₀²
    = −3 · 2π · 4
    = −24π
```

**Answer: ∮_C (y³ dx − x³ dy) = −24π** (C anticlockwise).

## Check yourself

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

1. Green's theorem: contour(P dx+Q dy) =
   A. (P_x - Q_y)
   B. (P_y - Q_x)
   C. double integral of (Q_x - P_y)
   D. (Q_x + P_y)
   Answer: C. Standard. (Key point: Green)

2. Q_x - P_y for P=y^3,Q=-x^3:
   A. 3(x^2+y^2)
   B. -3x^2+3y^2
   C. 0
   D. -3(x^2+y^2)
   Answer: D. -3x^2 - 3y^2. (Key point: Integrand)

3. Integral becomes:
   A. -3 * integral r^2 * r dr dtheta
   B. -3 integral r dr dtheta
   C. 3 integral r^2 dr dtheta
   D. -3 integral r^3 dr
   Answer: A. Include Jacobian r. (Key point: Polar)

4. The answer:
   A. 24 pi
   B. -24 pi
   C. -12 pi
   D. -8 pi
   Answer: B. -3*2pi*4. (Key point: Result)

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