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

# Apply Green's theorem

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

- Asked in: 2080 Bhadra, 1+4 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

State Green's theorem in plane and apply it to evaluate ∮_C [(2x − y + 4)dx + (5y + 3x − 6)dy] around a triangle in the xy-plane with vertices at (0,0), (3,0), (3,2).

## Short answer

∂Q/∂x − ∂P/∂y = 4. The double integral over the triangle equals 4 × area = 4 × ½ × 3 × 2 = 12.

## Answer

#### Green's theorem in the plane
If C is a simple closed curve (anticlockwise) bounding a region R, and P(x,y), Q(x,y) have continuous first partial derivatives in R, then

∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy.

#### Application
```text
P = 2x − y + 4,   Q = 5y + 3x − 6
∂Q/∂x = 3,   ∂P/∂y = −1
∂Q/∂x − ∂P/∂y = 3 + 1 = 4
```

The triangle has vertices (0,0), (3,0), (3,2). The hypotenuse is y = 2x/3. So x runs from 0 to 3 and y from 0 to 2x/3.

```text
∮_C = ∬_R 4 dx dy = 4 ∫₀³ ∫₀^(2x/3) dy dx
    = 4 ∫₀³ (2x/3) dx
    = 4 · (2/3) · (9/2)
    = 12
```

Check: area of triangle = ½ · 3 · 2 = 3, and 4 × 3 = 12.

**Answer: ∮_C = 12.**

## Check yourself

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

1. Green's theorem relates?
   A. Flux to volume integral
   B. Line integral around a closed curve to a double integral of dQ/dx - dP/dy
   C. Gradient to potential
   D. Curl to divergence
   Answer: B. It converts a closed line integral to an area integral. (Key point: Statement)

2. For P = 2x-y+4, Q = 5y+3x-6?
   A. 2
   B. 3
   C. dQ/dx - dP/dy = 4
   D. -4
   Answer: C. 3 - (-1) = 4. (Key point: Integrand)

3. The triangle (0,0),(3,0),(3,2) has hypotenuse?
   A. y = 3x/2
   B. y = 2x
   C. y = x
   D. y = 2x/3
   Answer: D. Line through origin and (3,2). (Key point: Region)

4. Value?
   A. 12
   B. 6
   C. 3
   D. 24
   Answer: A. 4 x area 3. (Key point: Result)

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