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Trapezoidal rule

The trapezoidal rule approximates $\int_a^b f(x)\,dx$ by replacing the curve with straight segments between the points, so the area becomes a sum of trapezoids. Halving the step size $h$ roughly quarters the error, since the global error is $O(h^2)$ .

\int_a^b f\,dx \approx h\Big(\tfrac{f_0}{2} + f_1 + \cdots + f_{n-1} + \tfrac{f_n}{2}\Big)

the trapezoidal rule with

n

intervals

Symbols

$h$	step size, $(b-a)/n$
$f_i$	function value at point $i$

Example

$\int_0^3 x^2\,dx$ with 2 intervals, $h = 1.5$ :

$h(f_0/2 + f_1 + f_2/2) = 10.125$ (exact is 9).

The trapezoidal rule overestimates the area for a convex (upward-curving) function.

Practise interpolation and integration for free →

← Fixed-point iteration · Simpson's rule →

Part of Numerical Methods: Interpolation and integration.