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

# Verify Green's theorem

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

- Asked in: 2083 Baishakh, 5 marks; 2075 Asoj, 5 marks (asked 2 times)
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Green's theorem in the plane for ∮_C [(3x² − 8y²)dx + (4y − 6xy)dy], where C is the region bounded by y = x² and x = y².

## Short answer

Compute ∂Q/∂x − ∂P/∂y = 10y. Integrate over y from x² to √x, x from 0 to 1 yields 3/2. The line integral along both curve segments also sums to 3/2, confirming Green’s theorem.

## Answer

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

Here P = 3x² − 8y², Q = 4y − 6xy. C is the closed curve bounded by y = x² and x = y², which meet at (0, 0) and (1, 1). Traverse C counter-clockwise: along y = x² from O(0, 0) to A(1, 1), then back along x = y² from A to O.

#### Right side: double integral
```text
∂Q/∂x = −6y ,   ∂P/∂y = −16y   ⇒   ∂Q/∂x − ∂P/∂y = 10y
```
For 0 ≤ x ≤ 1, y runs from x² to √x:

```text
∬ 10y dy dx = ∫₀¹ 5[y²]_{x²}^{√x} dx = 5 ∫₀¹ (x − x⁴) dx = 5(1/2 − 1/5) = 3/2
```

#### Left side: line integral
**Along y = x² (x: 0 → 1), dy = 2x dx:**
```text
∫ [(3x² − 8x⁴) + (4x² − 6x³)(2x)] dx = ∫₀¹ (3x² + 8x³ − 20x⁴) dx = 1 + 2 − 4 = −1
```

**Along x = y² (y: 1 → 0), dx = 2y dy:**
```text
∫ [(3y⁴ − 8y²)(2y) + (4y − 6y³)] dy = ∫₁⁰ (6y⁵ − 22y³ + 4y) dy
 = −[ y⁶ − 11y⁴/2 + 2y² ]₀¹ = −(1 − 5.5 + 2) = 5/2
```

```text
∮_C = −1 + 5/2 = 3/2
```

**Both sides equal 3/2, 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 = 3x^2 - 8y^2 and Q = 4y - 6xy, what is dQ/dx - dP/dy?
   A. 10y
   B. -22y
   C. -10y
   D. 2y
   Answer: A. dQ/dx = -6y and dP/dy = -16y, so the difference is -6y + 16y = 10y. (Key point: Green's integrand)

2. For the region between y = x^2 and x = y^2, what are the limits of the double integral?
   A. 0 <= x <= 1, sqrt(x) <= y <= x^2
   B. 0 <= x <= 1, x^2 <= y <= sqrt(x)
   C. 0 <= x <= 2, x^2 <= y <= sqrt(x)
   D. 0 <= x <= 1, 0 <= y <= 1
   Answer: B. The curves meet at (0,0) and (1,1); between them sqrt(x) lies above x^2. (Key point: Region limits)

3. What does 'verify' Green's theorem require?
   A. Compute only the double integral
   B. Show the line integral is zero
   C. Compute the line integral and the double integral separately and show both equal 3/2
   D. Compute the line integral along y = x^2 only
   Answer: C. Verification means evaluating both sides independently and showing they match. (Key point: Verification meaning)

4. How is C traversed for the line integral?
   A. Clockwise: O to A along x = y^2, then back along y = x^2
   B. Both curves from O to A
   C. Only along y = x^2
   D. Counter-clockwise: O to A along y = x^2, then A to O along x = y^2
   Answer: D. Green's theorem uses the positive (counter-clockwise) orientation. (Key point: Direction of traversal)

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