Trapezoidal Rule

Summary

The trapezoidal rule approximates the area under $y = f (x)$ by trapezoids. The composite rule is second-order accurate for smooth $f$ .

With $h = (b - a) / n$ , $x_{i} = a + i h$ , $y_{i} = f (x_{i})$ ,

\int_{a}^{b} f (x) d x \approx \frac{h}{2} (y_{0} + 2 y_{1} + \dots + 2 y_{n - 1} + y_{n}) .

If $f^{″}$ is continuous,

E = - \frac{(b - a)}{12} h^{2} f^{″} (ξ)

for some $ξ \in (a, b)$ . Thus the error is $O (h^{2})$ .

$\int_{0}^{1} x^{2} d x$ with $n = 2$ , $h = 1 / 2$ :

\frac{1 / 2}{2} (0 + 2 \cdot (1 / 2)^{2} + 1) = \frac{1}{4} \cdot \frac{3}{2} = \frac{3}{8} = 0.375 .

Exact: $1 / 3 \approx 0.333$ .

Claiming a specific absolute accuracy from a large $n$ without using the error formula or a comparison integral.
Applying the smooth-error formula across singularities.

![[TrapezoidalRule.mp4]]

Composite trapezoidal quadrature is standard.^[1]