Trapezoid Integral

Numerical integration by the trapezoidal rule

The trapezoidal rule approximates ∫f(x)dx by replacing the area under f with a chain of trapezoids. Dividing [a,b] into n subintervals of width h = (b−a)/n gives ∫ ≈ h/2 · (f(x₀) + 2·Σf(xᵢ) + f(xₙ)). The error is O(h²) — halving when you double n. Simpson's 1/3 rule fits parabolas instead and reaches O(h⁴) for the same cost per point; Romberg integration applies Richardson extrapolation on top of trapezoidal results and accelerates convergence further. Example: ∫₀¹ x² dx = 1/3. With n=4 trapezoids you get ≈ 0.3438; with n=100, 0.33335; with n=10000, essentially the exact value.

Applications and context

Trapezoidal integration is everywhere: volume and area under curves in CAD, energy consumed (integral of power over time), rainfall analysis from hydrograph curves, consumer/producer surplus in economics, signal-processing RMS and regression diagnostics. Whenever you have tabulated data without an analytic formula — sensor logs, financial flows — it's usually the first integrator you reach for.

FAQ

How do I choose n? Start small, double n, compare results: if they agree to your tolerance, stop. This adaptive refinement is the basis of Romberg.

Trapezoidal vs Simpson — when is Simpson worth it? Whenever f is smooth, Simpson gives orders of magnitude more accuracy for the same number of evaluations. For noisy or piecewise data, trapezoidal is more robust.

What about non-uniform spacing? The trapezoidal rule generalizes: each trapezoid uses its own width (xᵢ₊₁−xᵢ)·(f(xᵢ)+f(xᵢ₊₁))/2. Perfect for irregular sensor samples.