Simpson's Rule Calculator

Approximate the definite integral of any function over [a, b] using composite Simpson's 1/3 rule with a user-defined even number of subintervals.

Use the Simpson's Rule Calculator

Inputs

Supports: +, -, *, /, ^, sin, cos, tan, exp, log, ln, sqrt, abs, pi, e

Must be even — odd values are rounded up automatically.

Enter values above and click Calculate to see the approximation.

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Summary

Simpson's Rule Calculator approximates the definite integral of a function over [a, b] by fitting parabolic arcs between successive triplets of points. The composite Simpson's 1/3 rule divides the interval into n equal subintervals (n must be even) and applies the formula: (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ⋯ + 4f(xₙ₋₁) + f(xₙ)], where h = (b−a)/n. Because it uses quadratic interpolation, it is more accurate than the trapezoidal rule for the same n and exact for polynomials up to degree 3.

How it works

  1. Enter a mathematical function such as x^2, sin(x), or e^x in the function field.
  2. Set the lower bound a and upper bound b of the integration interval.
  3. Choose an even number of subintervals n (higher values give greater accuracy).
  4. Click Calculate to apply composite Simpson's 1/3 rule.
  5. Review the approximation result, the step-size h, and the full partition table with coefficient weights.

Use cases

  • Estimate areas under curves when an exact antiderivative is hard to find.
  • Verify calculus homework using Simpson's rule approximations.
  • Compare accuracy against the trapezoidal rule for the same interval.
  • Integrate empirical data or physics functions numerically.
  • Teach or learn the composite Simpson's 1/3 rule step by step.
  • Check numerical integration results before applying higher-order methods.
  • Explore how error shrinks as n grows — Simpson's error is O(h^4).

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu