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

# Evaluate a line integral

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

- Asked in: 2083 Baishakh, 5 marks
- Video: [Evaluating Line Integrals](https://www.youtube.com/watch?v=Tz14rC0XvHI) — Professor Dave Explains
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate the line integral ∫_C [(x² + y²) i − 2xy j]·dr, where C is the parabola x² = y from (0,0) to (1,1).

## Short answer

Parameterize the parabola as y = x^2, dy = 2x dx, x from 0 to 1. The line integral becomes ∫_0^1 (x^2 - 3x^4) dx = -4/15.

## Answer

Let F = (x² + y²) i − 2xy j and dr = dx i + dy j + dz k, so

```text
F·dr = (x² + y²) dx − 2xy dy
```

Along C: y = x², dy = 2x dx, with x from 0 to 1.

```text
∫_C F·dr = ∫₀¹ [(x² + x⁴) − 2x(x²)(2x)] dx
         = ∫₀¹ (x² + x⁴ − 4x⁴) dx
         = ∫₀¹ (x² − 3x⁴) dx
         = [x³/3 − 3x⁵/5]₀¹
         = 1/3 − 3/5
         = (5 − 9)/15
         = −4/15
```

**Answer: ∫_C F·dr = −4/15.**

## Check yourself

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

1. F.dr for F = (x^2+y^2) i - 2xy j is?
   A. (x^2+y^2) dx + 2xy dy
   B. x^2 dx - y^2 dy
   C. (x^2 + y^2) dx - 2xy dy
   D. (x^2+y^2) dy
   Answer: C. Dot product with dx i + dy j. (Key point: Integrand)

2. On y = x^2, what is dy?
   A. x dx
   B. 2 dx
   C. x^2 dx
   D. 2x dx
   Answer: D. Differentiate y = x^2. (Key point: Substitution)

3. After substitution the integrand becomes?
   A. x^2 - 3x^4
   B. x^2 + 5x^4
   C. x^2 - x^4
   D. 3x^4
   Answer: A. x^2 + x^4 - 4x^4 = x^2 - 3x^4. (Key point: Simplified integrand)

4. The value is?
   A. 4/15
   B. -4/15
   C. 1/3
   D. -2/15
   Answer: B. 1/3 - 3/5 = -4/15. (Key point: Result)

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