---
title: "Derive Simpson's 1/3 rule"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:derive-simpson-13"
source: https://www.bctengineer.com/q/semester_4/numerical_method/derive-simpson-13
site: BCT Engineering
---

# Derive Simpson's 1/3 rule

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2069 Bhadra, 4 marks
- Video: [Chapter 07.03: Lesson: Simpson's One Third Rule: Derivation](https://www.youtube.com/watch?v=qAVk2P5-Nvc) — numericalmethodsguy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/derive-simpson-13
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

Derive the expression of Simpson's 1/3 rule for integration.

## Short answer

Simpson's 1/3 rule fits a parabola through three equally spaced points and integrates it exactly. The result for one pair of intervals is (h/3)(y0 + 4y1 + y2); the composite form uses weights 1,4,2,4,...,1 with error -(b-a)h^4 f''''(xi)/180.

## Answer

Simpson's 1/3 rule replaces f(x) on two sub-intervals by a parabola (2nd-degree interpolating polynomial) through three equally spaced points and integrates it exactly.

#### Derivation
Take points x₀, x₁ = x₀ + h, x₂ = x₀ + 2h with values y₀, y₁, y₂. Put p = (x − x₀)/h, so x = x₀ + ph and dx = h dp. Newton's forward interpolation up to second differences gives

```text
f(x) ≈ y₀ + p·Δy₀ + p(p−1)/2 · Δ²y₀
```

Integrate from x₀ to x₂, i.e. p = 0 to 2:

```text
∫ f dx = h ∫₀² [ y₀ + pΔy₀ + (p²−p)/2 · Δ²y₀ ] dp
       = h [ y₀·p + Δy₀·p²/2 + Δ²y₀·(p³/6 − p²/4) ]₀²
       = h [ 2y₀ + 2Δy₀ + Δ²y₀·(8/6 − 1) ]
       = h [ 2y₀ + 2Δy₀ + (1/3)Δ²y₀ ]
```

Use Δy₀ = y₁ − y₀ and Δ²y₀ = y₂ − 2y₁ + y₀:

```text
∫ f dx = h [ 2y₀ + 2(y₁ − y₀) + (1/3)(y₂ − 2y₁ + y₀) ]
       = (h/3) [ 6y₀ + 6y₁ − 6y₀ + y₂ − 2y₁ + y₀ ]
       = (h/3) [ y₀ + 4y₁ + y₂ ]
```

This is Simpson's 1/3 rule for one pair of sub-intervals. Its error is −(h⁵/90) f⁗(ξ).

#### Composite rule
For n (even) sub-intervals, h = (b − a)/n, adding the n/2 pairs: the interior odd-indexed points appear once with weight 4 and the interior even-indexed points appear twice (end of one pair, start of next) with weight 2:

```text
∫ₐᵇ f dx ≈ (h/3) [ y₀ + yₙ + 4(y₁ + y₃ + … + yₙ₋₁) + 2(y₂ + y₄ + … + yₙ₋₂) ]
```

Total error ≈ −(b−a)h⁴/180 · f⁗(ξ), so the rule is exact for polynomials up to degree 3.

## Check yourself

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

1. What does Simpson's 1/3 rule do over two sub-intervals?
   A. Fits a straight line through two points
   B. Fits a cubic through four points
   C. Averages three function values
   D. Fits a parabola through three equally spaced points and integrates it
   Answer: D. It integrates the 2nd-degree Newton forward polynomial. (Key point: Idea)

2. After p = (x - x0)/h, what are the limits of integration?
   A. p = 0 to 2
   B. p = 0 to 1
   C. p = -1 to 1
   D. p = 0 to 3
   Answer: A. x0 to x2 covers two intervals, i.e. p from 0 to 2. (Key point: Limits)

3. What does the integral of the Δ²y0 term give?
   A. (1/2)Δ²y0 times h
   B. (1/3)Δ²y0 times h
   C. (2/3)Δ²y0 times h
   D. zero
   Answer: B. ∫0² (p²-p)/2 dp = (8/3 - 2)/2 = 1/3. (Key point: Integral of the Δ² term)

4. Up to what polynomial degree is Simpson's rule exact?
   A. Degree 2 only
   B. Degree 4
   C. Degree 3
   D. Degree 1
   Answer: C. The error involves f'''' so cubics are integrated exactly. (Key point: Exactness)

## Related in Numerical Method

- [Composite Simpson 1/3 algorithm](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Asimpson-13-algorithm) — Numerical Method, chapter 5
- [Simpson 1/3: 4e^x/(1+x³)](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Asimpson-13-4e-x) — Numerical Method, chapter 5
- [Simpson 3/8 pseudo-code](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Asimpson-38-pseudocode) — Numerical Method, chapter 5
- [Derive linear least squares](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aleast-squares-linear-derive) — Numerical Method, chapter 4

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