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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\R$ be continuous.\nThe *trapezoidal rule* is the [[quadrature rule|ADSVMCY77A]]\n$$T(f) = \\frac{b - a}{2} \\big( f(a) + f(b) \\big).$$\nFor an integer $N \\ge 1$, let $h = (b - a)/N$ and $x_i = a + i h$ for $i = 0, \\dots, N$.\nThe *composite trapezoidal rule* with $N$ subintervals is\n$$T_N(f) = h \\left( \\frac{f(x_0)}{2} + f(x_1) + \\cdots + f(x_{N-1}) + \\frac{f(x_N)}{2} \\right),$$\nthe sum of $T$ over the subintervals $[x_{i-1}, x_i]$.\n\n## Remarks\n\n$T$ is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with $n = 1$, and its [[degree of exactness|ZYT1JNT1Y0]] is $1$.\nExample: 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 $\\int_0^1 x^2\\, dx = \\tfrac{1}{3}$.\nNon-example: the left Riemann sum $h \\big( f(x_0) + \\cdots + f(x_{N-1}) \\big)$ is not the composite trapezoidal rule.\nIt 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 $1/2$.\nThe composite rule is the average of the left and right Riemann sums.\nFor 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 $h$.\n",
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