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

# Evaluate a line integral

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

- Asked in: 2082 Kartik (new course), 2 marks
- 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∮_C e^z/((z − 1)(z − 2)) dz over the circle C: |z| = 3.

## Short answer

Both poles z=1 and z=2 lie inside |z|=3. Residues are -e and e^2, sum e^2 - e. By the residue theorem, the integral equals 2πi (e^2 - e) = 2πi e(e-1).

## Answer

Let f(z) = e^z/((z − 1)(z − 2)). The poles z = 1 and z = 2 are both inside C: |z| = 3. By Cauchy's residue theorem, ∮_C f dz = 2πi × (sum of residues).

```text
Res at z = 1:  e^z/(z − 2) at z = 1 = e/(−1) = −e
Res at z = 2:  e^z/(z − 1) at z = 2 = e²/1 = e²
Sum = e² − e
```

**Answer: ∮_C f(z) dz = 2πi (e² − e) = 2πi e (e − 1).**

## Check yourself

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

1. Which poles lie inside |z|=3?
   A. Only z=1
   B. Only z=2
   C. None
   D. Both z=1 and z=2
   Answer: D. Both have modulus less than 3. (Key point: Poles inside)

2. Residue at z=1:
   A. -e
   B. e
   C. e^2
   D. -e^2
   Answer: A. e^1/(1-2) = -e. (Key point: Residue at 1)

3. The integral is:
   A. 2 pi i (e - e^2)
   B. 2 pi i (e^2 - e)
   C. pi i (e^2 - e)
   D. 2 pi i e^2
   Answer: B. 2 pi i times sum of residues. (Key point: Result)

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