Simpson Rule Calculator

Work out simpson rule instantly with clear inputs, formula shown and shareable results.

About Simpson Rule Calculator

Simpson's rule fits a parabola through each pair of intervals, weighting the ordinates 1, 4, 2, 4, …, 4, 1. It is exact for cubics and its error falls as h⁴, far faster than the trapezoid rule's h².

Formula

Simpson's rule

∫ ≈ (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + … + 4fₙ₋₁ + fₙ]

Worked example

With the default values below, Simpson Rule Calculator returns Simpson's rule estimate of 6.2000325521. Change any input above and every figure here is recalculated in your browser.

Inputs used

Polynomial coefficients, highest power first: 1, 0, 0, 0, 0
From x =: 1
To x =: 2
Number of intervals (even)accepts 2–10,000: 8

Results

Simpson's rule estimate: 6.2000325521
Exact integral: 6.2000000000
Absolute error: 0.000032552083
Trapezoid rule for comparison: 6.2364501953

Step by step

1Values used
2Simpson's rule
3Simpson's rule estimate= 6.2000325521
4Exact integral= 6.2000000000
5Absolute error= 0.000032552083
6Trapezoid rule for comparison= 6.2364501953

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 12 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Integrate x⁴ from 0 to 2 with 8 intervals.

The exact value is 32/5 = 6.4 and Simpson gives 6.4000651 — an error of only 0.00007.

Why must the interval count be even?

Each parabola spans two intervals, so an odd count would leave one unpaired.