Lab · Engineering Mathematics

Numerical Integration Lab

Integrate one definite integral with the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature at once, refine each of them, and read accuracy against the number of function evaluations it cost.

Problem

f(x)

Operators + − * / ^, parentheses, pi and e, and sin, cos, tan, asin, acos, atan, exp, sqrt, abs, ln, log, log10. Write 2*x, not 2x.

Lower bound a[–]

Upper bound b[–]

Reference integral I_ref (optional)

Errors below are measured against this value, which is taken exactly as entered.

Methods and refinement

Composite trapezoidal[–]

Composite Simpson (even n)[–]

Gauss-Legendre (max order)[–]

Refinement levels[–]

Composite rules double n at each level; Gauss-Legendre steps up to the chosen maximum order. A composite base setting is validated before any sequence is built: a fractional n, or an odd n for Simpson, is reported and nothing is run. One sweep may spend at most 40,000 evaluations in total; each level reserves its worst-case cost before f is touched, so a level that fails late still consumed the work it did and the cap can never be exceeded. A zero-width interval costs nothing at all and reserves nothing. Drawing the sample points of the final successful configuration re-evaluates that one configuration again, outside this cap, for display only.

Presets

Comparison

Method	Setting	Estimate Q	Evaluations	Absolute error \|Q − I_ref\|	Relative error \|Q − I_ref\| / \|I_ref\|
Composite trapezoidal	n = 8	1.974231602	9	2.577e-2	1.288e-2
	n = 16	1.993570344	17	6.430e-3	3.215e-3
	n = 32	1.998393361	33	1.607e-3	8.033e-4
	n = 64	1.999598389	65	4.016e-4	2.008e-4
Composite Simpson	n = 8	2.00026917	9	2.692e-4	1.346e-4
	n = 16	2.000016591	17	1.659e-5	8.296e-6
	n = 32	2.000001033	33	1.033e-6	5.167e-7
	n = 64	2.000000065	65	6.453e-8	3.227e-8
Gauss-Legendre	m = 2	1.935819575	2	6.418e-2	3.209e-2
	m = 3	2.001388914	3	1.389e-3	6.945e-4
	m = 4	1.999984228	4	1.577e-5	7.886e-6
	m = 5	2.00000011	5	1.103e-7	5.514e-8

Absolute error is |Q − I_ref|; the relative error is the same quantity divided by |I_ref|, a dimensionless ratio.

Error against cost

Composite trapezoidalComposite SimpsonGauss-Legendre

Absolute error against the supplied reference integral, on a logarithmic vertical axis, plotted against a linear count of the function evaluations each level cost. A level costing exactly zero evaluations is representable here; an exactly zero error cannot be drawn on the logarithmic axis and is read from the table.

Integrand and sample points

f(x)interval bounds a, bComposite trapezoidal · n = 64 (markers thinned)Composite Simpson · n = 64 (markers thinned)Gauss-Legendre · m = 5

f(x) = sin(x) on [0, 3.142], with the sample points of the final successful configuration of every enabled method. A method with no successful level contributes no nodes.

Every node was used in the sums; a dense partition draws an evenly spaced subset of markers for legibility only. Marker thinning never changes a numerical result, and reconstructing these nodes for the plot is not counted as quadrature cost.

What this Lab models

One proper definite integral of a scalar function over a finite interval, attacked by three rules of very different character: a second-order composite rule, a fourth-order composite rule, and a rule that places its points optimally. The Lab exists to make the trade-off visible — accuracy is read against the number of times f had to be evaluated, which is the only cost that matters when f is expensive.

Model scope

· Finite interval, proper integral, scalar integrand. Infinite bounds and unbounded integrands are out of scope.
· a > b is integrated in the orientation given and returns the correct negative value; a = b is exactly zero.
· A non-finite value at any required node is reported as a failure for that level, never skipped.
· Composite rules refine by doubling n; Gauss-Legendre refines by raising the order, up to 8.
· Simpson requires an even n at every level and rejects an odd one rather than rounding it.
· With no reference integral, successive differences are reported as refinement differences — a convergence indicator, not an error.

Related calculations

Reading the presets

· sin(x) · [0, π] — ∫₀^π sin(x) dx = 2 exactly. A smooth, well-scaled benchmark: Simpson should beat the trapezoidal rule by orders of magnitude at the same cost, and a low-order Gauss rule beats both.
· exp(x) · [0, 1] — ∫₀¹ eˣ dx = e − 1 ≈ 1.718281828459045. Analytic everywhere, so every nominal convergence order is genuinely attained.
· 1/(1+x²) · [0, 1] — ∫₀¹ dx/(1+x²) = π/4 ≈ 0.7853981633974483. Smooth but with a nearby complex singularity, which slows Gauss convergence compared with eˣ.
· sqrt(x) · [0, 1] — ∫₀¹ √x dx = 2/3 ≈ 0.6666666666666666. Finite, but the derivative blows up at x = 0: the integrand is not smooth there, so the O(h²) and O(h⁴) error statements no longer apply and the observed rates fall short.