What is Simpson's rule?

Simpson's 1/3 rule is a numerical integration method that approximates the area under a curve by fitting parabolic arcs through consecutive triplets of points. The composite version applies this over n subintervals, giving the formula (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ⋯ + f(xₙ)].

Simpson's 1/3 rule processes subintervals in pairs — each pair of consecutive subintervals forms one parabolic segment. An odd n would leave a dangling unpaired subinterval with no matching formula, so n must be even.

How accurate is Simpson's rule?

The truncation error is proportional to h^4 (where h = (b−a)/n), so doubling n reduces the error by a factor of roughly 16. It is significantly more accurate than the trapezoidal rule (error ∝ h²) for the same n, and it integrates polynomials up to degree 3 exactly.

What functions can I enter?

You can use standard math expressions: arithmetic (+, -, *, /, ^), trig functions (sin, cos, tan, asin, acos, atan), exponentials (exp, e^x), logarithms (log, ln), sqrt, abs, and constants pi and e.

How does it differ from the trapezoidal rule?

The trapezoidal rule uses straight-line (linear) segments between points, while Simpson's rule uses parabolic (quadratic) arcs. Simpson's rule achieves higher accuracy — typically 4th-order vs 2nd-order — but n must be even.

Simpson's Rule Calculator

Name: Simpson's Rule Calculator
Availability: InStock
Author: Nham Vu

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.

Inputs

f(x) — Function

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

Lower bound a

Upper bound b

Subintervals n (even, 2 – 1000)

Must be even — odd values are rounded up automatically.

Partition Table

i	x_i	f(x_i)	Weight

Enter values above and click Calculate to see the approximation.

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Summary

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.

How it works

Enter a mathematical function such as x^2, sin(x), or e^x in the function field.
Set the lower bound a and upper bound b of the integration interval.
Choose an even number of subintervals n (higher values give greater accuracy).
Click Calculate to apply composite Simpson's 1/3 rule.
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: 2026-06-13 · Reviewed by Nham Vu