---
title: "Evaluate a line integral F=(x−y)i+(x+y)j"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-f-x"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-x
site: BCT Engineering
---

# Evaluate a line integral F=(x−y)i+(x+y)j

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

- Asked in: 2069 Chaitra, 5 marks; 2068 Chaitra, 5 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-x
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∮_C F·dr, where F = (x − y) i + (x + y) j along the closed curve C bounded by y² = x and x² = y.

## Short answer

Using Green's theorem, ∂Q/∂x − ∂P/∂y = 2. The area between y²=x and x²=y from x=0 to 1 is 1/3. The line integral equals 2 × 1/3 = 2/3.

## Answer

F = (x − y) i + (x + y) j, so P = x − y and Q = x + y. The closed curve C is bounded by y² = x and x² = y, which meet at O(0, 0) and A(1, 1).

#### By Green's theorem
```text
∮_C F·dr = ∬_R (∂Q/∂x − ∂P/∂y) dx dy = ∬_R (1 − (−1)) dx dy = 2 · Area(R)
```

Area between the curves (for 0 ≤ x ≤ 1, y runs from x² to √x):

```text
Area = ∫₀¹ (√x − x²) dx = 2/3 − 1/3 = 1/3
```

```text
∮_C F·dr = 2 × 1/3 = 2/3
```

#### Direct check
- Along y = x² (x: 0 → 1), dy = 2x dx: ∫ [(x − x²) + (x + x²)2x] dx = ∫₀¹ (x + x² + 2x³) dx = 1/2 + 1/3 + 1/2 = 4/3.
- Along x = y² (y: 1 → 0), dx = 2y dy: ∫ [(y² − y)2y + (y² + y)] dy = ∫₁⁰ (2y³ − y² + y) dy = −(1/2 − 1/3 + 1/2) = −2/3.

Total = 4/3 − 2/3 = 2/3 ✓.

**Answer: ∮_C F·dr = 2/3** (counter-clockwise).

## 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, what is dQ/dx - dP/dy?
   A. 0
   B. 1
   C. -2
   D. 2
   Answer: D. 1 - (-1) = 2. (Key point: Green's integrand)

2. The line integral equals what?
   A. 2 times the area of the region
   B. The area
   C. Zero
   D. The perimeter
   Answer: A. Integrand is constant 2. (Key point: Reduction)

3. What is the area between y = x^2 and y = sqrt(x)?
   A. 2/3
   B. 1/3
   C. 1
   D. 1/2
   Answer: B. Integral of sqrt(x) - x^2 from 0 to 1 = 2/3 - 1/3. (Key point: Area)

4. What is the value?
   A. 1/3
   B. 4/3
   C. 2/3
   D. -2/3
   Answer: C. 2 x 1/3 = 2/3. (Key point: Result)

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