Simpson's Rule Calculator
Approximate definite integrals using Simpson's Rule with comprehensive step-by-step solutions and error analysis. Our calculus calculator supports numerical integration, accuracy comparison, detailed calculations, and educational explanations for advanced mathematics.
Last updated: December 15, 2024
Need a custom calculus calculator for your educational platform? Get a Quote
Enter function using standard notation: x^2, sin(x), cos(x), ln(x), e^x, etc.
Must be even for Simpson's Rule (will be adjusted if odd)
Integral:
∫[0 to 1] (x^2) dx
Integration Analysis
Simpson's Rule Approximation:
0.333333
Analysis Type:
Step-by-step solution
Exact Value:
0.333333
Error Analysis:
Absolute: 0.000000, Relative: 0.0000%
Formula:
∫[0 to 1] (x^2) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)]
Solution Steps:
- 1. Verify that n is even (if not, increase by 1)
- 2. Calculate step size: h = (b - a)/n = (1 - 0)/4 = 0.250000
- 3. Apply Simpson's Rule coefficients: 1, 4, 2, 4, 2, ..., 4, 1
- 4. Calculate f(xi) for each point
- 5. Multiply f(xi) by corresponding coefficient
- 6. Sum all products
- 7. Multiply by h/3 = 0.083333
- 8. Final approximation: 0.333333
Calculation Table:
| i | x_i | f(x_i) | Coeff | Product |
|---|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 1 | 0.0000 |
| 1 | 0.2500 | 0.0625 | 4 | 0.2500 |
| 2 | 0.5000 | 0.2500 | 2 | 0.5000 |
| 3 | 0.7500 | 0.5625 | 4 | 2.2500 |
| 4 | 1.0000 | 1.0000 | 1 | 1.0000 |
Analysis:
Simpson's Rule with 4 intervals gives ∫[0 to 1] (x^2) dx ≈ 0.333333. The exact value is 0.333333.
Simpson's Rule Tips:
- • Formula: (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)]
- • Pattern: Coefficients are 1, 4, 2, 4, 2, ..., 4, 1
- • Accuracy: Error is O(h⁴), very accurate for smooth functions
- • Requirement: Number of intervals must be even
Quick Example Result
For ∫₀¹ x² dx using Simpson's Rule with 4 intervals:
≈ 0.333333
Exact value: 1/3 ≈ 0.333333 (Error: 0.0000%)
How This Calculator Works
Our Simpson's Rule calculator applies the parabolic approximation method for numerical integration. This technique divides the integration interval into even subintervals and fits parabolas through consecutive point triplets, providing highly accurate approximationswith O(h⁴) error convergence for smooth functions.
Simpson's Rule Algorithm
Ensure n is even (adjust if necessary)h = (b - a) / nPattern: 1, 4, 2, 4, 2, ..., 4, 1∫ ≈ (h/3) × Σ[coefficient × f(xi)]Simpson's Rule algorithm systematically applies parabolic interpolation across the integration domain. The coefficient pattern 1-4-2-4-2-...-4-1 emerges from integrating parabolas over pairs of subintervals. This method achieves O(h⁴) accuracy by exactly integrating cubic polynomials, making it highly effective for smooth functions where higher-order terms contribute minimally to the error.
Mathematical foundation for parabolic numerical integration
Mathematical Foundation
Simpson's Rule is based on Newton-Cotes quadrature formulas and uses parabolic interpolation to approximate the integrand. By fitting parabolas through sets of three consecutive points, the method captures the curvature of the function much better than linear approximations. The theoretical foundation relies on Lagrange interpolation and the fact that integrating a parabola over two subintervals yields the characteristic 1-4-1 weighting pattern. The method's high accuracy comes from its ability to integrate polynomials up to degree 3 exactly.
- Parabolic approximation captures function curvature better than linear methods
- O(h⁴) error convergence provides rapid accuracy improvement with smaller step sizes
- Exact integration for cubic polynomials makes it highly effective for smooth functions
- Coefficient pattern emerges naturally from integrating interpolating parabolas
Sources & References
- Numerical Analysis - Richard L. Burden and J. Douglas FairesComprehensive treatment of numerical integration methods and error analysis
- Wolfram MathWorld - Simpson's Rule ReferenceDetailed mathematical reference with derivations and examples
- Calculus: Early Transcendentals - James Stewart, Daniel K. Clegg, Saleem WatsonEducational treatment of numerical integration and approximation methods
Need help with other integration methods? Check out our trapezoidal rule calculator and integration calculator.
Get Custom Calculator for Your PlatformExample Analysis
Problem Setup:
- Function: f(x) = e^x
- Bounds: [0, 1]
- Intervals: n = 6
- Step size: h = 1/6 ≈ 0.1667
Coefficient Pattern:
Result: Simpson's approximation ≈ 1.718282, Exact value = e - 1 ≈ 1.718282
This example demonstrates Simpson's Rule's exceptional accuracy for smooth functions like e^x. The calculation follows the pattern: ∫₀¹ e^x dx ≈ (h/3)[1·e^0 + 4·e^(1/6) + 2·e^(2/6) + 4·e^(3/6) + 2·e^(4/6) + 4·e^(5/6) + 1·e^1]. With h = 1/6, this gives (1/18)[1·1.0000 + 4·1.1816 + 2·1.3956 + 4·1.6487 + 2·1.9477 + 4·2.3009 + 1·2.7183] = (1/18)·30.9293 ≈ 1.718282. The exact value is e - 1 ≈ 1.718282, showing remarkable accuracy even with just 6 intervals. The error is approximately 10⁻⁶, demonstrating the O(h⁴) convergence rate that makes Simpson's Rule so effective for numerical integration of smooth functions.
Frequently Asked Questions
Found This Calculator Helpful?
Share it with others who need help with numerical integration and calculus
Suggested hashtags: #Calculus #SimpsonRule #Math #NumericalIntegration #Calculator