The plainest way to find the area under a curve: slice it into strips and top each with a straight line connecting its two ends — a trapezoid. Add the strip areas. It’s less accurate than [[the-simpson-rule|Simpson’s]] curved tops, but it’s dead simple, always works, and halving the strip width quarters the error (O(h²)). It’s the workhorse first integral. Slide to add strips and watch it close on the true area.
The trapezoidal rule approximates ∫ᵏᵇf by connecting successive sample points with straight lines, summing trapezoid areas: (h/2)[f₀+2f₁+2f₂+…+fₙ]. Its error is O(h²) — halving h quarters the error — making it simpler but lower-order than [[the-simpson-rule|Simpson’s]] O(h⁴). On a convex function it OVER-estimates (chords sit above the curve). It is exact for straight lines, the basis of the Euler–Maclaurin formula, and (composited/adaptive) a default quadrature. A fail-loud self-check throws unless ∫₀³x² approaches 9 within 1e−3 for many strips. ◆ real numerical methods, node-verified.
Lower-order than Simpson (O(h²) vs O(h⁴)) and biased on curved functions (over on convex, under on concave). The chord areas and O(h²) error are exact.