---
title: "Romberg: e^(-x²) from 1 to 2"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:romberg-e-x2-1-2"
source: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-e-x2-1-2
site: BCT Engineering
---

# Romberg: e^(-x²) from 1 to 2

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2069 Bhadra, 6 marks
- Video: [Numerical Integration Part 13: Romberg Integration](https://www.youtube.com/watch?v=_BgF_b1MFSU) — Shree Ram Khaitu
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-e-x2-1-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

Evaluate ∫ from 1 to 2 of e^(-x²) dx using Romberg method correct up to 3 decimal places.

## Short answer

Build trapezoidal values for h = 1, 0.5, 0.25, 0.125 and apply Richardson extrapolation. The successive results agree to three decimals, giving the integral approximately 0.135.

## Answer

Romberg integration: build trapezoidal values T(h) for h = 1, 0.5, 0.25, 0.125, then extrapolate with R(i,j) = [4ʲ·R(i,j−1) − R(i−1,j−1)] / (4ʲ − 1), until two successive values agree to 3 decimals.

f(x) = e^(−x²), a = 1, b = 2.

#### Romberg table (6 decimals kept for accuracy)
| h | T (order h²) | R1 | R2 | R3 |
|---|---|---|---|---|
| 1 | 0.193098 |  |  |  |
| 0.5 | 0.149248 | 0.134632 |  |  |
| 0.25 | 0.138720 | 0.135210 | 0.135249 |  |
| 0.125 | 0.136121 | 0.135254 | 0.135257 | 0.135257 |

Check: R2 at h = 0.25 is 0.135249 and R3 at h = 0.125 is 0.135257; both round to 0.135, so the required 3-decimal accuracy is reached.

**Answer: ∫₁² e^(−x²) dx ≈ 0.135** (exact value 0.135257).

## Check yourself

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

1. Which h values are used?
   A. 1, 2, 3, 4
   B. 1, 0.9, 0.8, 0.7
   C. 1, 0.5, 0.25, 0.125
   D. 0.5, 0.5, 0.5, 0.5
   Answer: C. h is halved each row. (Key point: Step halving)

2. What is T(h=1) for [1, 2]?
   A. f(1.5) = 0.105
   B. f(1)+f(2) = 0.386
   C. 0.135257
   D. (1/2)[f(1) + f(2)] = 0.193098
   Answer: D. T(1) = (h/2)(f(a)+f(b)). (Key point: First trapezoid)

3. Why can the table stop at R3 with 0.135?
   A. Successive extrapolated values agree to 3 decimals
   B. All T values are equal
   C. h reached 0.125
   D. R1 equals T
   Answer: A. R2 = 0.135249 and R3 = 0.135257 both round to 0.135. (Key point: Stop)

4. How do T and R columns compare?
   A. T is more accurate than R
   B. T converges slowly; extrapolated values converge faster
   C. All columns are equal
   D. R values diverge
   Answer: B. Richardson extrapolation raises the order. (Key point: Convergence)

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