---
title: "Work done by a force field F=(2x−y+2z)i+(x+y−z)j…"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:work-done-by-a-force-field-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/work-done-by-a-force-field-two
site: BCT Engineering
---

# Work done by a force field F=(2x−y+2z)i+(x+y−z)j…

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

- Asked in: 2071 Chaitra, 5 marks
- Video: [Line Integral Solved Problem: Finding Work Done Along a Straight Line Path by team education](https://www.youtube.com/watch?v=uRkW9kqwa-A) — TEAM Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/work-done-by-a-force-field-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Using the line integral, compute the work done by the force F = (2x − y + 2z) i + (x + y − z) j + (3x − 2y − 5z) k when it moves once around the circle x² + y² = 4, z = 0.

## Short answer

On the circle z=0, parametrize x=2cos t, y=2sin t. Then F·dr = (4−4 sin t cos t) dt, and work = ∫₀^{2π} (4−2 sin 2t) dt = 8π. Alternatively, by Stokes, curl F·k = 2, area = 4π, giving 8π.

## Answer

Work = ∮_C F·dr around the circle x² + y² = 4, z = 0. Evaluate it as a line integral.

Parametrise: x = 2cos t, y = 2sin t, z = 0, 0 ≤ t ≤ 2π. Then dx = −2sin t dt, dy = 2cos t dt, dz = 0.

```text
F·dr = (2x − y + 2z) dx + (x + y − z) dy + (3x − 2y − 5z) dz
     = (2x − y) dx + (x + y) dy       (z = 0, dz = 0)
     = (4cos t − 2sin t)(−2sin t) dt + (2cos t + 2sin t)(2cos t) dt
     = (−8 sin t cos t + 4 sin²t + 4 cos²t + 4 sin t cos t) dt
     = (4 − 4 sin t cos t) dt
W = ∫₀^2π (4 − 2 sin 2t) dt = 8π − 0 = 8π
```

**Check by Stokes' theorem:** (curl F)·k = ∂Q/∂x − ∂P/∂y = 1 − (−1) = 2. Area of circle = π(2)² = 4π, so W = 2 × 4π = 8π.

**Answer: Work done = 8π ≈ 25.13 units** (anticlockwise).

## Check yourself

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

1. On z = 0, F.dr?
   A. F.dz
   B. (2x-y)dx + (x+y)dy
   C. (x+y)dx
   D. 0
   Answer: B. dz=0. (Key point: Reduction)

2. After substitution?
   A. 8 sin t
   B. 4
   C. 4 - 4 sin t cos t
   D. 0
   Answer: C. Computed. (Key point: Integrand)

3. (curl F).k?
   A. 1
   B. 0
   C. 3
   D. 2
   Answer: D. 1-(-1). (Key point: Stokes check)

4. Work?
   A. 8pi
   B. 4pi
   C. 16pi
   D. 0
   Answer: A. 2 x 4pi. (Key point: Result)

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