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A Simpson's rule calculator (or Simpson's 1/3 rule calculator) approximates definite integrals using Simpson's approximation. Use this free Simpson's approximation calculator to perform numerical integration with step-by-step solutions. This Simpson method calculator uses parabolic interpolation for high-accuracy results. Also known as simpsons rule calc or simpson calculator, it's ideal for calculus and numerical analysis.
Last updated: February 2, 2026
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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
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:
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:
For ∫₀¹ x² dx using Simpson's Rule with 4 intervals:
≈ 0.333333
Exact value: 1/3 ≈ 0.333333 (Error: 0.0000%)
A Simpson's 1/3 rule calculator (or Simpson 1/3 rule calculator) is another name for Simpson's Rule calculator. The "1/3" refers to the h/3 factor in the formula: ∫[a to b] f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)]. This Simpson's 1/3 rule calculator uses parabolic approximation for high-accuracy numerical integration.
A Simpson's approximation calculator uses Simpson's Rule to approximate definite integrals numerically. It fits parabolas through sets of three consecutive points and integrates these parabolas to estimate the area under a curve. A Simpson method calculator implements this parabolic interpolation method, providing step-by-step solutions, error analysis, and detailed calculation tables. Also known as simpsons rule calc or simpson calculator, it's ideal for calculus and numerical analysis.
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.
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
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.
Need help with other integration methods? Check out our trapezoidal rule calculator and integration calculator.
Get Custom Calculator for Your PlatformResult: 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.
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