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Simpson's Rule Calculator

Estimate a definite integral using the composite Simpson's rule.

Examples

Simpson's rule is exact for polynomials up to degree 3, so the first two examples return the exact answer.

Simpson's rule needs an even number of subintervals, so an odd n is rounded up to the next even number.

Supported syntax

Operators: + - * / % ^ and parentheses

Constants: pi, e

Functions: sqrt, cbrt, abs, exp, ln, log, log10, log2, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, floor, ceil, round, sign, pow

Important: write x^2, not x2. The letter x is the only variable.

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What Is Simpson's Rule Calculator?

The Simpson Rule Calculator estimates a definite integral with the composite Simpson rule, and reports the step size, a refined estimate taken with twice as many subintervals, and the difference between the two as an error estimate. You type the integrand using x, set the two limits and the number of subintervals, and the function is evaluated across the interval automatically. Five worked examples ship with the tool, each with a known exact value you can compare against.

Who Uses the Simpson's Rule Calculator?

Engineering and science students use it to check a numerical integration answer against a known result, and engineers use it for a quick estimate of area, volume, work or average value before committing to a full simulation.

How to Use the Simpson's Rule Calculator

  1. Type the integrand into the Function f(x) box using x, for example x^2 and not x2, then set the Lower limit a and the Upper limit b.
  2. Choose the number of Subintervals n, which must be at least 2, and press Estimate Integral. An odd count is rounded up to the next even number, which the rule requires.
  3. Read the estimate with the step size h, the refined 2n estimate and the error estimate, or click one of the example buttons to fill the form and see the comparison against a known exact value.

Why Choose Our Simpson's Rule Calculator?

  • The refined 2n estimate and the error figure let you judge convergence, so you can see whether n is large enough instead of guessing
  • Examples with known answers act as a self test: x^2 from 0 to 3 gives 9, sin(x) from 0 to pi gives 2, exp(x) from 0 to 1 gives e - 1, 1/x from 1 to e gives 1 and x^3 - 2x + 1 from 0 to 2 gives 4/3
  • The expression parser handles sqrt, cbrt, abs, exp, the logarithms, the trigonometric and hyperbolic functions, floor, ceil, round, sign and pow, plus the constants pi and e
  • Exact for polynomials up to degree 3, which is why the cubic examples return 9 and 4/3 to full precision rather than an approximation

Try the Simpson's Rule Calculator above — it is free, fast, and works on any device. Bookmark this page to return whenever you need it.