Simpson's Rule Calculator
Simpson's rule estimates a definite integral by fitting parabolas through pairs of subintervals, giving far better accuracy than the trapezoidal rule for smooth functions.
Function f(x)
Simpson estimate of the area under the curve
How does this calculator work?
Simpson's rule approximates ∫ₐᵇ f(x) dx as (Δx/3)[f(x₀) + 4·Σ odd + 2·Σ even + f(xₙ)] with Δx = (b − a)/n and even n. By fitting parabolas it is exact for cubics and has Δx⁴ error, so it beats the trapezoidal rule for smooth functions.
Formula
How this is calculated
Pick a function f(x), the lower limit a, the upper limit b, and the number of subintervals n. Simpson's rule requires n to be even because it groups the interval into pairs of strips, each capped by a parabola; if you enter an odd n the calculator increments it to the next even value.
The interval [a, b] is divided into n equal pieces of width Δx = (b − a)/n, giving sample points xᵢ = a + i·Δx for i = 0…n. The integral is approximated by (Δx/3) times the weighted sum f(x₀) + f(xₙ) plus 4 times every odd-indexed interior point and 2 times every even-indexed interior point. The 4-2-4-2 weighting reproduces the exact integral of the parabola through each triple of points.
Simpson's rule is exact for polynomials up to degree three and its error shrinks like Δx⁴, so doubling n cuts the error roughly sixteen-fold. Results assume f is continuous on [a, b]; singularities such as 1/x at x = 0 or ln(x) at x ≤ 0 produce non-finite samples and no result. Limits may be given in either order.
Frequently asked questions
Simpson's rule fits one parabola across each pair of adjacent subintervals, so the number of subintervals must be divisible by two. An odd n is automatically rounded up to the next even number.
It is exact for cubic and lower-degree polynomials and its error is proportional to Δx⁴, making it much more accurate than the trapezoidal or midpoint rules for the same number of points.
If any sample point produces a non-finite value (for example 1/x at x = 0), no approximation is returned. Choose an interval where the function is continuous.
Also known as
TG we-Calculate Editorial Team. (2026). Simpson's Rule Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/simpsons-rule-calculator
TG we-Calculate Editorial Team. "Simpson's Rule Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/simpsons-rule-calculator.
TG we-Calculate Editorial Team, "Simpson's Rule Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/simpsons-rule-calculator
@misc{wecalculate_simpsons_rule_calculator, title = {Simpson's Rule Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/simpsons-rule-calculator}}, year = {2026}, note = {TG we-Calculate} }
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