---
title: "Evaluate a line integral F=(x²+y²)i−2xyj"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-f-x2"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-x2
site: BCT Engineering
---

# Evaluate a line integral F=(x²+y²)i−2xyj

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

- Asked in: 2076 Chaitra, 5 marks
- Video: [If f= (x^2+y^2)i-2xyj evaluate integral c F.dr  the curve is rectangle in xy- plane y=0,y=b ,x=0,x=a](https://www.youtube.com/watch?v=OSV-zN7JcjE) — Sudhakar Maths academy
- 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-x2
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∮_C F·dr, where C is the rectangle bounded by the lines x = ±a, y = 0, y = b and F = (x² + y²) i − 2xy j.

## Short answer

Using Green's theorem, ∂Q/∂x − ∂P/∂y = −4y. The double integral over the rectangle −a≤x≤a, 0≤y≤b gives −4ab², so ∮_C F·dr = −4ab².

## Answer

Let P = x² + y², Q = −2xy. C is the closed rectangle −a ≤ x ≤ a, 0 ≤ y ≤ b, taken anticlockwise. Use Green's theorem:

```text
∮_C F·dr = ∬_R (∂Q/∂x − ∂P/∂y) dx dy
∂Q/∂x = −2y,   ∂P/∂y = 2y
∂Q/∂x − ∂P/∂y = −4y

∬_R (−4y) dx dy = ∫₋ₐ^a ∫₀^b (−4y) dy dx
                = ∫₋ₐ^a (−2b²) dx
                = −2b² · 2a
                = −4ab²
```

**Direct check** (anticlockwise: bottom, right, top, left):

```text
Bottom y = 0, x: −a→a:   ∫ x² dx = 2a³/3
Right  x = a, y: 0→b:    ∫ −2ay dy = −ab²
Top    y = b, x: a→−a:   ∫ (x² + b²) dx = −(2a³/3 + 2ab²)
Left   x = −a, y: b→0:   ∫ 2ay dy (since −2xy = 2ay) = −ab²
Sum = 2a³/3 − ab² − 2a³/3 − 2ab² − ab² = −4ab²
```

**Answer: ∮_C F·dr = −4ab²** (anticlockwise; reverse the sign for clockwise).

## Check yourself

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

1. For P = x^2 + y^2, Q = -2xy, dQ/dx - dP/dy?
   A. 0
   B. -2y
   C. 4y
   D. -4y
   Answer: D. -2y - 2y. (Key point: Integrand)

2. Rectangle?
   A. -a <= x <= a, 0 <= y <= b
   B. 0 <= x <= a
   C. -a <= x <= a, -b <= y <= b
   D. 0 <= y <= a
   Answer: A. As given. (Key point: Region)

3. Value?
   A. 4ab^2
   B. -4ab^2
   C. -2ab^2
   D. 0
   Answer: B. (-2b^2)(2a). (Key point: Result)

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