Last updated: 2026-07-01

Simpson's rule

Simpson's rule estimates a definite integral by replacing the curve with parabolas rather than straight line segments. Each pair of adjacent strips is fitted with a parabola through three points, and the integral is approximated by the total area under those parabolas. Because a parabola tracks curvature that a straight chord misses, Simpson's rule is usually far more accurate than the trapezoidal rule for the same sampling effort.

Split the interval [a, b] into n equal subintervals of width h = (b − a) / n, where n is even. This creates n + 1 grid points x₀ = a, x₁, x₂, …, xₙ = b. Sampling the function at each point gives the values f(x₀), f(x₁), …, f(xₙ), and the estimate is:

∫ₐᵇ f(x) dx ≈ (h/3)·[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)]

This calculator works from sampled function values rather than a typed formula: you provide f at the n + 1 evenly spaced points, which is exactly the situation you face when the data comes from measurements or a table.

Why the weights are 1, 4, 2, 4, 2, …, 4, 1

Take a single block of three points [x₀, x₁, x₂] with spacing h. Fit the unique parabola through them and integrate it exactly over the block; the result is

(h/3)·(f₀ + 4f₁ + f₂)

so the block midpoint carries weight 4 and the two ends weight 1. Now join consecutive blocks — [x₀, x₁, x₂], [x₂, x₃, x₄], and so on. Every block midpoint keeps its weight of 4, while each shared boundary point belongs to two blocks and picks up weight 1 + 1 = 2. The two outer endpoints belong to a single block and stay at weight 1. Collecting terms produces the alternating pattern 1, 4, 2, 4, 2, …, 4, 1, all multiplied by h/3. The pairing of blocks is exactly why n must be even.

Worked example

Estimate ∫₀¹ x² dx using n = 4 subintervals. The step size is h = (1 − 0) / 4 = 0.25, and the grid points are 0, 0.25, 0.5, 0.75, 1. The sampled values of x² are:

Apply the rule with weights 1, 4, 2, 4, 1:

∫ ≈ (0.25/3)·[0 + 4(0.0625) + 2(0.25) + 4(0.5625) + 1] ∫ ≈ (0.25/3)·[0 + 0.25 + 0.5 + 2.25 + 1] = (0.25/3)·4 = 1/3

The result is exactly 1/3, which is the true value of the integral. This is not luck: Simpson's rule integrates any polynomial up to degree three exactly, so a quadratic like x² is captured with zero error regardless of how many strips are used.

How the error behaves

Simpson's rule is fourth-order accurate: halving h cuts the error by about a factor of sixteen. For a function with a continuous fourth derivative, the error over [a, b] is

E = −((b − a)·h⁴/180)·f⁽⁴⁾(ξ)

for some point ξ in the interval. Two consequences follow:

• Because the error depends on the fourth derivative, any polynomial of degree three or less is integrated exactly — its fourth derivative is zero.
• For smooth curves the h⁴ factor drives the error down very quickly, so a modest number of strips often suffices for high accuracy.

Simpson's rule versus the trapezoidal rule

The trapezoidal rule joins points with straight segments, so its error is proportional to h² (second-order). Simpson's rule fits parabolas and reaches h⁴ (fourth-order), converging far faster for smooth functions at the same number of points. The trade-offs:

• Simpson's rule requires an even number of subintervals, whereas the trapezoidal rule works with any n.
• For noisy or irregular data, forcing parabolas through scattered points can amplify the noise; the trapezoidal rule is sometimes more robust there.

A practical rule of thumb: reach for Simpson's rule whenever the function is smooth and you want high accuracy for the same sampling effort; use the trapezoidal rule for quick estimates, noisy data, or when n must be odd.

Practical notes

• Even subintervals only. Simpson's 1/3 rule pairs the strips, so n must be even. The calculator flags an odd n.
• Match the count. With n subintervals you must supply exactly n + 1 values, one per grid point. The calculator flags a mismatch so the spacing stays uniform.
• Keep the order. List values from x₀ = a to xₙ = b. Interior odd- and even-indexed points are weighted differently, so the order matters.
• Reversed limits. If a is greater than b, the step size is negative and the result changes sign — matching the convention that swapping integration limits negates the integral.
• Equal spacing only. This is the uniform-grid Simpson's rule. Unequally spaced data needs a more general quadrature formula.

Related calculators

• Definite Integral of Polynomial Calculator — exact integral of a polynomial via the fundamental theorem of calculus
• Polynomial Derivative Calculator — numerical derivative at a single point
• Arithmetic Sequence Calculator — terms and sums of evenly spaced values