Trapezoidal Rule

Definition

The trapezoidal rule is a numerical integration method that approximates ${\int }_a^{b}f(x)dx$ by summing the areas of trapezoids formed between consecutive function evaluations.

Intuition

Rather than approximating the function by a flat horizontal line (rectangle rule), the trapezoidal rule connects neighbouring sample points with straight-line segments. Each trapezoid captures the slope of the function locally, giving a better approximation with the same number of evaluations.

Formal Description

Single interval $[0,h]$: Approximate $f$ by the line through $(0,f(0))$ and $(h,f(h))$:

$${\int }_0^{h}f(x)dx\approx \frac{h}{2}(f(0)+f(h)).$$

Composite rule on $[a,b]$: Divide into $n$ subintervals of width $h=\frac{b-a}{n}$, with nodes $f_k=f(a+kh)$:

$${\int }_a^{b}f(x)dx\approx \frac{h}{2}(f_0+2f_1+\cdots +2f_{n-1}+f_n).$$

All interior evaluations are multiplied by 2; the result is scaled by $\frac{h}{2}$.

Applications

Used whenever the antiderivative of $f$ is unknown or impractical to compute analytically — e.g. integrating tabulated experimental data or numerically solving ODEs (trapezoidal method for stiff problems). It is the simplest baseline for adaptive quadrature routines.

Trade-offs

The global error is $O(h^2)$: halving the step size quarters the error. Simpson’s rule achieves $O(h^4)$ at the same cost by fitting parabolas instead of lines, making it preferable for smooth functions. The trapezoidal rule is exact for linear functions and well-suited to periodic functions integrated over a full period (spectral convergence in that case).

Links

• Limits and Continuity — convergence of the approximation as $h\to 0$