Simpson's Rule

Simpson’s Rule is a method for numerical integration, specifically for estimating the area under a curve when a function is complex or difficult to integrate analytically. Named after Thomas Simpson, this rule is widely used in engineering, mathematics, and physics to calculate approximate values using parabolas.

Let’s explore the definition, formula, derivation, solved examples, and real-world uses of Simpson’s Rule.

 

Table of Contents

• What is Simpson’s Rule?
• Simpson’s Rule Formula
• Derivation of Simpson’s Rule
• Solved Examples
• Practice Questions
• Real-Life Applications
• Conclusion
• FAQs on Simpson’s Rule

 

What is Simpson’s Rule?

Simpson’s Rule is a technique to approximate the definite integral of a function by fitting second-degree polynomials (parabolas) to sections of the curve. It is most effective when the function is hard to integrate or when data is available in discrete points.

 

Simpson’s Rule Formula

The basic formula for Simpson’s 1/3 Rule is:

∫ₐᵇ f(x) dx ≈ (h/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]

Where:

• h = (b - a) / n is the width of each subinterval (n must be even)

• x₀, x₁, ..., xₙ are equally spaced points from a to b

• f(x) is the function to integrate
  
  

 

Derivation of Simpson’s Rule

To derive Simpson’s Rule:

• Take three points: (x₀, f(x₀)), (x₁, f(x₁)), (x₂, f(x₂))

• Fit a quadratic polynomial:
  f(x) = ax² + bx + c

• Integrate the polynomial over the interval [x₀, x₂]
  
  

The result simplifies to Simpson’s 1/3 Rule. The derivation confirms that Simpson’s Rule gives exact results for quadratic and cubic functions.

 

Solved Examples

Example 1:
Estimate ∫₀⁶ (x² + 1) dx using Simpson’s Rule with 4 intervals.

Solution:
n = 4, h = (6 - 0) / 4 = 1.5
x-values: 0, 1.5, 3, 4.5, 6
f(x) = x² + 1
→ f(0) = 1
→ f(1.5) = 3.25
→ f(3) = 10
→ f(4.5) = 21.25
→ f(6) = 37

Apply Simpson's Rule:
(1.5 / 3) × [1 + 4(3.25) + 2(10) + 4(21.25) + 37]
= 0.5 × [1 + 13 + 20 + 85 + 37]
= 0.5 × 156 = 78

Answer: Approximate integral = 78

 

Example 2:
Estimate ∫₁⁵ (1/x) dx using Simpson’s Rule with 4 intervals.

Solution:
h = (5 - 1) / 4 = 1
x-values: 1, 2, 3, 4, 5
f(x) = 1, 0.5, 0.333, 0.25, 0.2

Apply Simpson's Rule:
(1/3) × [1 + 4(0.5) + 2(0.333) + 4(0.25) + 0.2]
= (1/3) × [1 + 2 + 0.666 + 1 + 0.2]
= (1/3) × 4.866 ≈ 1.622

Answer: Approximate value ≈ 1.622

Practice Questions

1. Approximate ∫₀⁴ x³ dx using Simpson’s Rule with 4 intervals.

2. Compare Simpson’s Rule and Trapezoidal Rule results for ∫₁³ ln(x) dx.

3. Prove that Simpson’s Rule gives exact value for any quadratic function.
   
   

 

Real-Life Applications

• Engineering: Simulating stress, strain, and fluid flow

• Physics: Calculating work, energy, and displacement

• Economics: Estimating area under supply/demand curves

• Environmental Science: Measuring pollutant exposure, rainfall, etc.

 

Conclusion

Simpson’s Rule is a powerful numerical technique for approximating definite integrals, especially for functions that are difficult to integrate analytically. It is more accurate than the Trapezoidal Rule due to its parabolic interpolation, and it’s widely used in science, engineering, and data analysis.

 

Frequently Asked Questions on Simpson’s Rule

1. What is Simpson’s Rule used for?

Answer.It’s used to approximate definite integrals of complex or non-integrable functions.

2. Why must the number of intervals be even in Simpson’s Rule?

Answer.Because each parabolic segment covers two intervals, so the total must be even.

3. What’s the difference between Trapezoidal Rule and Simpson’s Rule?

Answer.Trapezoidal Rule uses linear approximations, while Simpson’s Rule uses parabolic curves for better accuracy.

4. Can Simpson’s Rule give exact results?

Answer.Yes, it gives exact results for all polynomials up to degree 3.

5. Who developed Simpson’s Rule?

Answer.Though named after Thomas Simpson, the method was known earlier by Johannes Kepler and others.

Want to simplify complex integrations and score higher in exams? Learn and master Simpson’s Rule with interactive visuals, examples, and expert tips at Orchids The International School.