---
title: "Evaluate a line integral F=zi+xj+yk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-f-zi"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-f-zi
site: BCT Engineering
---

# Evaluate a line integral F=zi+xj+yk

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

- Asked in: 2067 Magh
- Video: [Line Integral | To find Work Done | Numerical | Vector Calculus | Maths](https://www.youtube.com/watch?v=Tj7s06Gvc54) — Gautam Varde
- 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-zi
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∫_C F·dr where F = z i + x j + y k and C is the arc of the curve x = t² + 1, y = 2t², z = t³ from t = 1 to t = 2.

## Short answer

Parametrize: dx=2t dt, dy=4t dt, dz=3t² dt. F·dr = z dx + x dy + y dz = (8t⁴+4t³+4t) dt. Integrate from t=1 to 2: ∫(8t⁴+4t³+4t)dt = [8t⁵/5 + t⁴ + 2t²]₁² = 353/5 = 70.6.

## Answer

Here F·dr = z dx + x dy + y dz, and the curve is x = t² + 1, y = 2t², z = t³, with t from 1 to 2.

```text
dx = 2t dt,  dy = 4t dt,  dz = 3t² dt

z dx = t³ · 2t dt = 2t⁴ dt
x dy = (t² + 1)(4t) dt = (4t³ + 4t) dt
y dz = 2t² · 3t² dt = 6t⁴ dt

F·dr = (8t⁴ + 4t³ + 4t) dt
```

```text
∫₁² (8t⁴ + 4t³ + 4t) dt = [8t⁵/5 + t⁴ + 2t²]₁²
 = (256/5 + 16 + 8) − (8/5 + 1 + 2)
 = 248/5 + 21
 = 353/5
```

**Answer: ∫_C F·dr = 353/5 = 70.6.**

## Check yourself

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

1. F.dr is written as:
   A. x dx + y dy + z dz
   B. y dx + z dy + x dz
   C. z dx + x dy + y dz
   D. dx + dy + dz
   Answer: C. F = z i + x j + y k. (Key point: Reducing)

2. Substituting the curve, F.dr equals:
   A. (8t^4 + 4t^3) dt
   B. (2t^4 + 4t^3 + 6t^4) dt only
   C. (8t^3 + 4t^2 + 4t) dt
   D. (8t^4 + 4t^3 + 4t) dt
   Answer: D. 2t^4 + (4t^3+4t) + 6t^4. (Key point: Parametrize)

3. The integration limits are:
   A. t = 1 to 2
   B. x = 1 to 2
   C. t = 0 to 1
   D. t = 0 to 2
   Answer: A. The question gives t from 1 to 2. (Key point: Limits)

4. The value is:
   A. 248/5
   B. 353/5
   C. 71
   D. 256/5
   Answer: B. [8t^5/5 + t^4 + 2t^2] from 1 to 2 = 353/5. (Key point: Value)

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