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

# Verify Green's theorem

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

- Asked in: 2076 Chaitra, 5 marks
- Video: [GREEN'S THEOREM VERIFICATION. ENGINEERING MATHEMATICS-II. #ioe #engineering #maths](https://www.youtube.com/watch?v=iXxLNEyhfJ0) — Mr. Ghale 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-greens-theorem-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Green's theorem in plane for ∮_C [(x − y)dx + (x + y)dy] where C is the boundary of the region enclosed by y² = x and x² = y.

## Short answer

∂Q/∂x − ∂P/∂y = 2. The double integral over the region between y=√x and y=x² is 2∫₀¹(√x − x²)dx = 2/3. The line integral also equals 2/3, confirming Green's theorem.

## Answer

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

The curves y² = x and x² = y meet at O(0,0) and A(1,1). In 0 ≤ x ≤ 1 the curve y = √x lies above y = x².

#### Double integral
```text
∂Q/∂x − ∂P/∂y = 1 − (−1) = 2
∬_R 2 dx dy = 2 ∫₀¹ (√x − x²) dx = 2 (2/3 − 1/3) = 2/3
```

#### Line integral (anticlockwise: O to A along y = x², then A to O along y² = x)
On C₁: y = x², dy = 2x dx, x from 0 to 1. Then (x − y) dx + (x + y) dy = (x − x²) + (x + x²)(2x) = x + x² + 2x³ per dx.

```text
∫₀¹ (x + x² + 2x³) dx = 1/2 + 1/3 + 1/2 = 4/3

C₂: x = y², dx = 2y dy, y: 1→0
 (y² − y)(2y) + (y² + y) = 2y³ − 2y² + y² + y = 2y³ − y² + y
 ∫₁⁰ (2y³ − y² + y) dy = −(1/2 − 1/3 + 1/2) = −2/3

Total = 4/3 − 2/3 = 2/3
```

Both sides equal 2/3, so **Green's theorem is verified.**

## Check yourself

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

1. For P = x-y, Q = x+y?
   A. 0
   B. 1
   C. x+y
   D. 2
   Answer: D. 1 + 1. (Key point: Integrand)

2. Value?
   A. 2/3
   B. 1/3
   C. 4/3
   D. 2
   Answer: A. 2 x 1/3. (Key point: Double integral)

3. Along y = x^2 from O to A?
   A. 1/3
   B. 4/3
   C. 2/3
   D. -4/3
   Answer: B. 1/2 + 1/3 + 1/2. (Key point: Line integral)

4. Sum of both pieces?
   A. 4/3 + 2/3 = 2
   B. 4/3
   C. 4/3 - 2/3 = 2/3
   D. -2/3
   Answer: C. Return path gives -2/3. (Key point: Total)

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