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Simpson's Rule Calculator
Apply Simpson's 1/3 rule to approximate definite integrals with error analysis and step-by-step solution.
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Try an Example
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Basic Polynomial
Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.
Key values: x^2 · [0, 1] · Simpson's rule
Trigonometric Integral
Integrate sin(x) from 0 to pi. Exact answer: 2.
Key values: sin(x) · [0, pi] · Exact: 2
Area Under a Bell Curve
Approximate the Gaussian integral e^(-x^2) from -3 to 3.
Key values: e^(-x^2) · [-3, 3] · Gaussian
Simpson's 1/3 Rule
Simpson's rule approximates definite integrals by fitting parabolas (quadratic polynomials) through consecutive groups of three points, rather than using straight lines (trapezoidal) or flat tops (midpoint). This gives convergence — dramatically faster than the of trapezoidal or midpoint rules.
where and must be even.
Error Bound
The error depends on the fourth derivative of . This means Simpson's rule is exact for polynomials up to degree 3 (since their fourth derivative is zero).
Practical tip: Doubling reduces the error by a factor of 16 (since ). With just 10 subdivisions, Simpson's rule often achieves 8+ digits of accuracy for smooth functions.
Why Simpson's Rule Is the Default
Among the three numerical methods this calculator supports, Simpson's rule offers the best accuracy per function evaluation. For smooth functions, it typically matches the exact answer to machine precision with far fewer subdivisions than the alternatives.
- vs. Trapezoidal: Same number of function evaluations, but 4th-order convergence vs. 2nd-order. For , Simpson's error is roughly times smaller.
- vs. Midpoint: Midpoint avoids endpoint singularities, but for smooth functions Simpson's converges much faster.
When Not to Use Simpson's Rule
- Endpoint singularities: If is undefined at an endpoint (e.g., at 0), the midpoint rule is more robust since it avoids evaluating at endpoints.
- Highly oscillatory functions: Functions with rapid oscillations need very fine subdivisions regardless of method. Adaptive quadrature may be better.
- Odd number of intervals: Simpson's rule requires an even number of subdivisions. If forced to use an odd number, the trapezoidal rule works.
Frequently Asked Questions
What is Simpson's 1/3 rule?
Simpson's 1/3 rule approximates a definite integral by fitting parabolas through consecutive groups of three points. It achieves convergence, meaning doubling the number of subdivisions reduces the error by a factor of 16.
Why does Simpson's rule require an even number of subintervals?
Each parabolic segment spans two consecutive subintervals (three points). If is odd, the last subinterval has no partner to complete a parabola, so the method cannot be applied uniformly.
How accurate is Simpson's rule compared to the trapezoidal rule?
For the same number of function evaluations, Simpson's rule is dramatically more accurate for smooth functions. With subdivisions, Simpson's error is roughly times smaller than the trapezoidal rule's error.
Is Simpson's rule exact for polynomials?
Yes, Simpson's rule is exact for polynomials of degree 3 or less. The error bound depends on the fourth derivative of the integrand, which is zero for cubics and lower-degree polynomials.
When should I avoid using Simpson's rule?
Avoid Simpson's rule when the function has singularities at the integration endpoints (use the midpoint rule instead), when the function oscillates rapidly (consider adaptive quadrature), or when you are limited to an odd number of subintervals.
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