Simpson's Rule Calculator
Last updated: 7 August 2026
Reviewed by Gavin Meiring, Lead research and primary author ยท Doctoral Candidate (Corporate Governance) ยท Research and drafting assisted by AI
- Simpson's rule is named for Thomas Simpson, an 18th-century English mathematician who published it in 1743 โ but Simpson was largely self-taught and worked as a weaver before turning to mathematics.
- The rule was actually known earlier: Johannes Kepler used the same idea around 1615 to estimate the volume of wine barrels, so it is sometimes called Kepler's rule.
- Simpson's rule fits a parabola through three points at a time, and for smooth functions its error shrinks like 1/nโด โ fast enough that it beat every rival until computers arrived.
Simpson's Rule Calculator
Simpson's rule calculator approximates definite integrals by fitting parabolic arcs through successive groups of three points on a curve, giving significantly better accuracy than the trapezoidal rule for the same number of subintervals. It is used by calculus students, scientists, and engineers who need precise numerical integration. Enter your function, bounds, and an even number of subintervals to get your result.
How to Use the Simpson's Rule Calculator
- Enter your function f(x) in the Simpson's Rule Calculator input field. Set the lower bound a and upper bound b.
- Enter an even number of subintervals n. Simpson's rule requires n to be even.
- Click Calculate to evaluate the weighted sum and display the approximation.
- Review the node values and their weights (1, 4, 2, 4, 2, ..., 4, 1) in the working table.
The Formula
For n subintervals (n must be even) with width delta_x = (b - a) / n:
S = (delta_x / 3) * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)]
The weighting pattern is 1, 4, 2, 4, 2, ..., 4, 1. Interior points alternate between weights of 4 and 2, with 4 assigned to odd-indexed points and 2 to even-indexed interior points. The error in Simpson's rule is proportional to (b-a)^5 * f''''(max) / (180n^4), making it considerably more accurate than the trapezoidal rule.
Real-World Example
Approximate the integral of f(x) = sin(x) from x = 0 to x = pi using 4 subintervals.
- delta_x = pi / 4
- Node x values: 0, pi/4, pi/2, 3pi/4, pi
- Function values: f(0) = 0, f(pi/4) = 0.7071, f(pi/2) = 1, f(3pi/4) = 0.7071, f(pi) = 0
- Apply the formula: S = (pi/4)/3 * [0 + 4(0.7071) + 2(1) + 4(0.7071) + 0]
- S = (pi/12) * [0 + 2.8284 + 2 + 2.8284 + 0] = 0.261799 * 7.6569 = 2.004560
The true value is exactly 2. Simpson's rule reaches 2.004560 with only 4 subintervals, an error of 0.23%.
Why Simpson's Rule Outperforms the Trapezoidal Rule
The trapezoidal rule approximates the curve with straight lines within each subinterval, introducing error wherever curvature exists. Simpson's rule fits a parabola through three consecutive points, capturing both the slope and the curvature of the function. Since most smooth functions resemble parabolas locally, this is a much better fit. The error in Simpson's rule decreases as n^4 rather than n^2, meaning doubling the number of subintervals reduces the error by a factor of 16 rather than 4. For integrals of polynomial functions of degree three or less, Simpson's rule is exact.
Convergence as the Interval Count Rises
Simpson's rule is exact for polynomials up to the third degree, and its error falls quickly for anything smooth. The figures below hold the function and the limits fixed and change only the number of subintervals.
| Subintervals (n) | Nodes | Simpson's value | Error against the exact 2 |
|---|---|---|---|
| 2 | 3 | 2.094395 | 0.094395 |
| 4 | 5 | 2.004560 | 0.004560 |
| 8 | 9 | 2.000269 | 0.000269 |
| 16 | 17 | 2.000017 | 0.0000166 |
| 32 | 33 | 2.000001 | 0.0000010 |
f(x) = sin(x) on the interval from 0 to pi, where the exact integral is 2. Doubling n divides the error by between fourteen and twenty-one across these rows, closing on the theoretical factor of sixteen.
The practical reading is short. Four subintervals already give three correct decimals on this function and eight give five. Where the function is smooth, adding nodes is a cheap way to buy accuracy, and doubling n once is enough to see whether the answer has settled.
Where the Rule Stops Working
The rule fits a parabola through each pair of adjacent intervals, which is why n must be even. With an odd n the weighting pattern of 1, 4, 2, 4, ... 1 does not close, and the tool rejects the input rather than returning a distorted answer. The error term is proportional to (b - a)^5 times the largest fourth derivative in the interval, divided by 180 n^4. For sin(x) between 0 and pi the fourth derivative is bounded by 1, so the worst case at n = 4 is pi^5 divided by 180 times 256, which is 0.00664. The table shows 0.004560, inside that bound.
Two limits matter in use. A kink or a discontinuity breaks the smoothness the error term assumes, and the estimate can be badly wrong however large n becomes. A function that oscillates faster than the node spacing can be missed by the sampled values altogether, which is the failure mode that produces a confident wrong answer. Read the node table the tool prints alongside the summary figure.
Frequently Asked Questions
Why must n be even for Simpson's rule? Simpson's rule groups points into pairs of subintervals (three points per group) to fit each parabola. An even number of subintervals means an odd number of nodes, which allows the groups to tile the interval without overlap or gaps.
What is Simpson's 3/8 rule? Simpson's 3/8 rule is a variant that groups points into triples of subintervals (four points per parabola) and uses the weighting pattern 1, 3, 3, 1. It requires n to be a multiple of 3 and has similar accuracy to the standard Simpson's rule. The standard rule (sometimes called Simpson's 1/3 rule) is more commonly used.
When is Simpson's rule not much better than the trapezoidal rule? If the function has large fourth derivatives or discontinuities within the interval, the error bound for Simpson's rule may not be much smaller. In those cases, adaptive methods that use finer grids in difficult regions are more effective.
Can Simpson's rule be applied to tabulated data? Yes, as long as the data points are equally spaced. This makes it practical for integrating experimental measurements where you have readings at regular intervals but no analytical formula for the function.
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Extended Reference Notes
The notes below cover the broader context that informs how to use the Simpson's Rule Calculator well.
Typical Input Ranges
Most real-world uses of the Simpson's Rule Calculator fall into a middle band where the result is stable and useful. Very small inputs to the Simpson's Rule Calculator often round to zero or near-zero, and very large inputs amplify every rounding error in the calculation. The middle band, where the Simpson's Rule Calculator inputs are ordinary sizes, is where the tool is most reliable.
Assumptions Behind the Formula
The Simpson's Rule Calculator assumes the inputs stay fixed across the period or scenario being modelled. Rates move, values change, and fees appear, so treat the Simpson's Rule Calculator output as a clean reference and layer in the frictions your own situation adds.
Common Edge Cases
Three situations change the Simpson's Rule Calculator answer in ways the formula does not surface: boundary values near zero, rounding cascades across many steps, and unit mismatches between fields. When any of these apply, sanity-check the Simpson's Rule Calculator result against an independent estimate.
When to Revisit the Calculation
The Simpson's Rule Calculator output is only as current as its inputs, so re-run the calculation whenever a key value changes materially. A quarterly re-check of the Simpson's Rule Calculator suits personal planning; monthly suits active business or investment decisions.
Relationship to Other Tools
The Simpson's Rule Calculator shares inputs and outputs with the other tools in its category. If the same numbers feed several tools, capture them once and run each tool so the comparison stays consistent with the Simpson's Rule Calculator.
Practical Checklist Before Relying on the Result
Before acting on the Simpson's Rule Calculator output, run a short mental checklist: inputs in the right units, direction of the result matching intuition, and magnitude plausible. Each check takes seconds and catches the most common classes of Simpson's Rule Calculator error before they reach a decision.
Putting the Result to Work
A single Simpson's Rule Calculator run usually narrows the range of plausible answers rather than settling the question. Compare the Simpson's Rule Calculator result against a benchmark or a previous run, and ask what would have to change for the answer to flip a decision.
Sensitivity to Inputs
Some inputs move the Simpson's Rule Calculator result more than others; changing each by a small amount shows which ones matter. Spend the effort on the high-impact Simpson's Rule Calculator inputs and treat the low-impact ones as approximate.
A Note on Stale Inputs
A calculation is only as fresh as the inputs that feed it, so note the date the Simpson's Rule Calculator inputs were last refreshed. A six-month-old Simpson's Rule Calculator result can be as wrong as a wrong calculation when the underlying values have moved on.