SL 5.8—Trapezoidal rule

For equally spaced x-values with width h, the trapezoidal rule approximates ∫ₐᵇ f(x)dx by h/2[f(x₀)+2f(x₁)+⋯+2f(xₙ₋₁)+f(xₙ)]. It estimates signed area, not automatically total geometric area.

The endpoint ordinates have weight one and interior ordinates weight two. More strips usually reduce error for a smooth curve, but curvature and the number of strips still determine the approximation quality.

Using two strips for f(x)=x² on [0,2] gives h=1 and (1/2)[0+2(1)+4]=3, while the exact integral is 8/3. The difference is the discretisation error.

Do not use the rule with the wrong h or omit the doubled interior values. If the curve crosses the axis, signed integral and total area require different treatment.