Simpson's three-eighths rule calculator for definite integrals

It applies the composite 3/8 weighting pattern across the interval and requires the subinterval count to be divisible by three. The result includes the approximation, step size, node count, and weighted sum, making it useful both for a quick answer and for checking a numerical integration exercise or implementation.

Enter the function and interval clearly

Write the integrand as an expression in x, then provide finite lower and upper bounds. Explicit multiplication is required, so enter 2*x rather than 2x. The expression parser supports ordinary arithmetic, parentheses, powers with the ^ symbol, the constants pi and e, and common one-argument functions including sin, cos, tan, exp, log, log10, sqrt, and abs. This focused syntax keeps the calculation deterministic and avoids executing arbitrary code. For example, x^3 + 2*x over the interval from zero to three is valid. Reversing the bounds is also valid and changes the sign of the approximation, just as it does for an exact definite integral. Every sampled function value must be finite. If the expression has a singularity, an invalid square root, division by zero, or another undefined value at one of the nodes, the calculator stops with a specific input error instead of returning a misleading numeric result. Check parentheses and multiplication signs carefully when adapting textbook notation.

Choose a subinterval count divisible by three

The composite Simpson three-eighths rule divides the integration interval into equal subintervals. Its basic panel spans three subintervals and four nodes, so the total number of subintervals must be a positive multiple of three: 3, 6, 9, and so on. If n is the subinterval count, the calculator uses n + 1 nodes, including both endpoints. It computes the uniform step size h = (b - a) / n and applies endpoint weights of one. Interior nodes whose indices are divisible by three receive weight two, while the remaining interior nodes receive weight three. The final weighted sum is multiplied by 3h/8. Increasing n often improves an approximation for a sufficiently smooth function, but it is not a universal guarantee when the function oscillates sharply, contains discontinuities, or behaves badly near the interval. Compare results at successively larger valid counts such as 6, 12, and 24 to assess practical stability rather than trusting extra digits automatically. The accepted maximum keeps execution bounded.

Interpret and verify the returned calculation

The primary output is approximation, the signed estimate of the definite integral. The response also reports step_size, subintervals, node_count, and weighted_sum so you can audit the setup or reproduce the final multiplication independently. Simpson's three-eighths rule is exact for polynomials through degree three when arithmetic is exact, which makes cubic examples especially useful as checks. For other smooth functions, its error depends on higher derivatives and the selected step size. A plausible-looking decimal does not prove that the function was entered correctly or that the mesh is fine enough. Confirm the interval orientation, examine whether the function remains finite and smooth throughout the interval, and repeat the calculation with a larger multiple of three. You can also compare against an antiderivative when one is available, or against a different numerical method. The API uses the same deterministic solver as the browser calculation and costs $0.002 per request, so automated checks and interactive work follow the same arithmetic contract.