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Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.
Key values: x^2 · [0, 1] · Simpson's rule
Integrate sin(x) from 0 to pi. Exact answer: 2.
Key values: sin(x) · [0, pi] · Exact: 2
Approximate the Gaussian integral e^(-x^2) from -3 to 3.
Key values: e^(-x^2) · [-3, 3] · Gaussian
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.
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.
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.
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.
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.
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.
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.
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.