Composite parabolic quadrature with a visible weight audit
Simpson Rule Integration Calculator
Approximate a definite integral from a safe preset function or from equally spaced sampled values. The result separates endpoint, odd-index, and even-index contributions and verifies the required even number of subintervals. Use the displayed node ribbon to check every coefficient before transferring the estimate into homework, a lab note, or an engineering worksheet for reproducible numerical work.
Choose the input bench
Composite Simpson result
(0.25/3) × [0 + 4 + 4(5.25) + 2(3.50)] = 2.66666667
Weighted node ribbon
x0=0.0000, f=0.0000, w=1 | x1=0.2500, f=0.0625, w=4 | x2=0.5000, f=0.2500, w=2 | x3=0.7500, f=0.5625, w=4 | x4=1.0000, f=1.0000, w=2 | x5=1.2500, f=1.5625, w=4 | x6=1.5000, f=2.2500, w=2 | x7=1.7500, f=3.0625, w=4 | x8=2.0000, f=4.0000, w=1
Refinement difference ÷ 15: 0.00000000. This is a practical smooth-function error proxy, not a guaranteed bound.
What Simpson’s rule is approximating
A definite integral accumulates signed values of a function over an interval. When f(x) is above the x-axis, its contribution is positive; below the axis, its contribution is negative. Simpson’s rule approximates that integral by fitting quadratic behavior across pairs of equal subintervals rather than using only straight trapezoids.
For one panel from a to b with midpoint m and h = (b−a)/2, the elementary formula is h[f(a)+4f(m)+f(b)]/3. Composite Simpson’s rule divides a larger interval into an even number n of equal subintervals and applies the pattern continuously. The endpoint weights are 1, odd interior nodes receive 4, and even interior nodes receive 2.
The result is a signed integral, not automatically a geometric area. If a function crosses the axis, positive and negative regions can cancel. Geometric area requires locating sign changes and integrating the absolute value piecewise. The linked area tool may use a different convention, so name the quantity you need before comparing answers.
The weight pattern in four parts
Endpoints: weight 1
f(x0) and f(xn) each appear once. They bracket the full integration interval and are not part of either interior sum.
Odd nodes: weight 4
f(x1), f(x3), through f(xn−1) are the midpoints of each two-subinterval Simpson panel.
Even nodes: weight 2
f(x2), f(x4), through f(xn−2) are shared boundaries between neighboring panels, producing the repeating coefficient 2.
Scale: h/3
Add the weighted values and multiply by h/3, where h = (b−a)/n. Reversing bounds makes h negative and reverses the integral sign.
Worked example: integrate x² from 0 to 2
The default uses n = 8, so h = (2−0)/8 = 0.25. The nodes are 0, 0.25, 0.50, …, 2. Their squared values are 0, 0.0625, 0.25, 0.5625, 1, 1.5625, 2.25, 3.0625, and 4.
The odd-index sum is 0.0625 + 0.5625 + 1.5625 + 3.0625 = 5.25. The even interior sum is 0.25 + 1 + 2.25 = 3.50. Substitute: (0.25/3)[0 + 4 + 4(5.25) + 2(3.50)] = (0.25/3)×32 = 8/3, or approximately 2.66666667.
Simpson’s rule integrates polynomials through degree three exactly in exact arithmetic, so refining x² from n = 8 to n = 16 produces the same value apart from floating-point rounding. That does not mean every function is exact. For a smooth nonpolynomial function, the difference between n and 2n estimates provides a useful convergence signal.
Function mode and sampled-data mode
Safe preset functions
Function mode deliberately offers a small list rather than evaluating arbitrary text as code. This keeps the embedded module self-contained and avoids ambiguous expression parsing. Select a function, bounds, and even n; the calculator generates every node.
Equally spaced observations
Data mode accepts y-values only, ordered from x0 to xn, plus a positive spacing h. Nine y-values create eight subintervals. The x origin is irrelevant to the weighted sum when h is already known.
No refinement from sparse data
A genuine 2n refinement needs midpoint values that are not present in the original data. The calculator does not invent them by interpolation, so refinement and its error proxy are marked unavailable in data mode.
Unequal spacing is different
Composite Simpson 1/3 weights assume constant h. Timestamps or measurements at irregular x-values require interpolation, resampling with justification, or a quadrature method designed for nonuniform nodes.
Why n must be even
Each Simpson panel spans two subintervals and uses three nodes: left endpoint, midpoint, and right endpoint. An even n lets the entire interval be tiled by these two-subinterval panels. If n is odd, the repeating 1-4-2-4 pattern cannot end correctly with a single final endpoint.
Do not silently drop the last subinterval. Options include changing n, collecting another equally spaced value, applying Simpson’s 3/8 rule to a suitable last group, or using a trapezoid for a leftover interval with an explicit mixed-method report. This calculator requires even n so the displayed result remains pure composite Simpson 1/3.
In data mode, an even number of subintervals means an odd number of y-values. Three values are the minimum. The calculator rejects nonnumeric entries and nonpositive h before assigning weights.
Error behavior and convergence
For a sufficiently smooth function with a continuous fourth derivative, the composite error has order h⁴. Halving h can reduce the leading error by about 16. This motivates the difference/15 proxy: if Sn and S2n are in the asymptotic regime, the refined estimate’s leading error is approximately |S2n−Sn|/15.
This proxy is not a guaranteed bound. Discontinuities, corners, singularities, rapid oscillation, narrow peaks, overflow, or insufficient refinement can break the expected pattern. A small difference can also be misleading if both grids miss the same localized feature. Inspect the function and compare multiple refinements or use adaptive quadrature for consequential work.
The classic theoretical error includes the fourth derivative at an unknown point. If a defensible maximum of |f(4)(x)| on [a,b] is available, it can produce a bound proportional to (b−a)h⁴/180. This calculator avoids pretending it knows that derivative bound from sampled values alone.
Numerical and interpretation checks
Confirm units. Integrating velocity in miles per hour over hours gives miles; integrating a probability density over its variable gives probability; integrating power over time gives energy after unit conversion. The vertical and horizontal units multiply. A bare “square units” label is not universally correct.
Reverse-bound results should be negatives of forward-bound results. A constant function c should integrate to c(b−a). Linear and quadratic test functions are useful implementation checks. If all sampled y-values are nonnegative and h is positive, the Simpson estimate should not be negative.
Use the verified area under curve calculator when the application specifically frames geometric area, and the on-site Simpson’s rule integral calculator as a related published destination. This fresh module emphasizes weight auditing, tabulated data, and refinement diagnostics.
Frequently asked questions
Can Simpson’s rule return a negative value?
Yes. A definite integral is signed. A function below the axis contributes negatively, and reversed limits negate the result. Geometric area requires absolute values and sign-change handling.
Why are there n+1 nodes?
n subintervals have two endpoints plus n−1 interior boundaries, totaling n+1 nodes. With even n, that node count is odd.
Can I use unevenly spaced data?
Not with this composite 1/3 implementation. Enter y-values from equal spacing h. Unequal nodes need another method or a justified preprocessing step.
Does increasing n always improve the answer?
Often for smooth functions until floating-point and evaluation error matter, but not as an absolute guarantee. Discontinuities, singularities, oscillations, or missed peaks require diagnosis and possibly adaptive methods.
Why is x² exact with only two subintervals?
Simpson’s rule is exact for polynomials up through degree three in exact arithmetic. Its quadratic interpolant represents x² perfectly.
Can I type any formula?
No. Function mode uses explicit safe presets. For another function, calculate equally spaced y-values with trusted software and use data mode, or use a maintained numerical integration library.
References
The formula and error structure follow the U.S. National Institute of Standards and Technology Digital Library of Mathematical Functions, Section 3.5(ii). NIST gives the elementary and composite Simpson formulas, requires even n for the composite pattern, and states the fourth-derivative error term proportional to h⁴.
For a reproducible numerical record, save the function definition or original y-values, integration bounds, equal spacing, subinterval count, software precision, signed-integral convention, and every refinement result. Report enough digits to support the application without implying that all displayed decimals are accurate. When an analytic antiderivative is available, compare the numerical value with it as a check. When it is unavailable, test convergence across several even n values and inspect the function for discontinuities, singularities, oscillations, and narrow features. Independent software based on a different quadrature family can provide another useful comparison.
In experimental data, measurement uncertainty and sampling spacing may dominate quadrature truncation error. Refining a mathematical grid cannot recover physical observations that were never collected. Interpolating between noisy measurements also introduces modeling choices that should be stated. If the accumulated quantity controls medical dosing, engineering safety, financial reporting, or another consequential decision, use validated domain software and expert review. This browser calculator is intended for education, transparent hand-checking, and preliminary exploration; it is not a certification of a numerical model or its inputs.