Composite Simpson Calculator for Numerical Integration

Calculator Inputs

Use function mode for analytic expressions or table mode for equally spaced x,y data.

Input mode

Function mode Table mode

f(x)

Supported: +, -, *, /, ^, sin, cos, tan, exp, ln, log, sqrt, abs, min, max, pow.

Lower limit a

Upper limit b

Even subintervals n

Known exact value (optional)

Decimal places

Known exact value (optional)

Use the shared field below when table mode is active.

Shared exact value field

Decimal places

Equally spaced x,y pairs

Example format: 0, 0 then 0.5, 0.25 and so on. The x spacing must remain constant and the number of subintervals must be even.

Example Data Table

This example uses y = x² on [0, 2] with h = 0.5 and n = 4.

i	x_i	y_i	Coefficient	Weighted term
0	0	0	1	0
1	0.5	0.25	4	1
2	1	1	2	2
3	1.5	2.25	4	9
4	2	4	1	4

Simpson estimate = (0.5 / 3) × (0 + 4 + 4×(0.25 + 2.25) + 2×1) = 2.666667, which matches the exact integral of x².

Formula Used

For an even number of subintervals n, composite Simpson’s rule approximates the definite integral by fitting parabolas over adjacent pairs of intervals.

h = (b - a) / n ∫[a to b] f(x) dx ≈ (h / 3) × [ f(x0) + f(xn) + 4 × Σ f(xi) for odd i + 2 × Σ f(xi) for even i, 2 ≤ i ≤ n-2 ]

For tabulated data, replace f(x_i) with y_i. This rule requires equally spaced x values and an even number of subintervals.

In function mode, the page also computes a refined S(2n) estimate and reports the Richardson-style error estimate |S(2n) − S(n)| / 15 for smooth integrands.

How to Use This Calculator

Choose Function mode for a mathematical expression or Table mode for known x,y values.
If you use function mode, enter f(x), the lower limit a, the upper limit b, and a positive even value for n.
If you use table mode, paste one x,y pair per line. Keep all x values equally spaced and ensure the total subinterval count is even.
Optionally enter a known exact integral to compare absolute and relative error.
Set the number of decimal places, then press Calculate Composite Simpson.
Review the result summary, node table, weighted terms, and export the report as CSV or PDF.

FAQs

1. What does composite Simpson’s rule approximate?

It approximates a definite integral by combining several Simpson 1/3 segments over an interval. Each segment uses quadratic interpolation across two adjacent subintervals to estimate area.

2. Why must n be even?

Simpson’s 1/3 rule works on pairs of subintervals. Because each parabola spans two intervals, the total number of subintervals must be even for the composite method.

3. Can I use tabulated data instead of a formula?

Yes. Enter one x,y pair per line in table mode. The x values must be equally spaced, and the number of subintervals formed by the data must stay even.

4. What happens if my x values are not equally spaced?

The calculator stops and shows a validation message. Standard composite Simpson’s rule assumes uniform spacing, so unequal gaps break the weighting pattern and the formula no longer applies directly.

5. How is the error estimate shown in function mode?

The page recomputes the integral with 2n subintervals and estimates error using |S(2n) − S(n)| / 15. This refinement works best for smooth functions.

6. Which functions can I type into f(x)?

You can use x, constants like pi and e, arithmetic operators, powers, and functions such as sin, cos, tan, exp, ln, log, sqrt, abs, min, max, and pow.

7. When is Simpson’s rule especially accurate?

It performs very well for smooth functions with modest curvature changes. It is exact for polynomials up to degree three when arithmetic is exact and spacing is uniform.

8. What do the coefficients 1, 4, 2, 4, …, 1 mean?

They are the Simpson weights applied to endpoints, odd interior points, and even interior points. These weights create the quadratic interpolation pattern used in the final area estimate.