Trapezoidal rule

Remarks

$T$ is the closed Newton–Cotes formula with $n = 1$, and its degree of exactness is $1$. Example: for $f(x) = x^2$ on $[0, 1]$ and $N = 2$, $T_2(f) = \tfrac{1}{2}\big(0 + \tfrac{1}{4} + \tfrac{1}{2}\big) = \tfrac{3}{8}$, while $\int_0^1 x^2\, dx = \tfrac{1}{3}$. Non-example: the left Riemann sum $h \big( f(x_0) + \cdots + f(x_{N-1}) \big)$ is not the composite trapezoidal rule. It gives full weight to $f(x_0)$ and none to $f(x_N)$; for $f(x) = x$ on $[0, 1]$ and $N = 1$ it gives $0$ instead of $1/2$. The composite rule is the average of the left and right Riemann sums. For a continuous function of period $b - a$, $T_N(f) = h \sum_{i=0}^{N-1} f(x_i)$, and for smooth periodic $f$ the error decreases faster than any power of $h$.