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  "body": "\n## Summary\n\nThe trapezoidal rule approximates $\\int_a^b f$ by the area of the trapezoid under the chord from $(a, f(a))$ to $(b, f(b))$.\nIts composite form splits $[a, b]$ into $N$ equal parts and adds the trapezoids, with error of order $h^2$ for smooth $f$, where $h = (b - a)/N$.\nIt is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with two nodes, and it is remarkably accurate for smooth periodic functions over a full period.\n\n## Explanations\n\n### L1 human\n\nReplace the curve over each small piece by the straight line between its endpoints, and add up the areas of the resulting trapezoids.\nEach trapezoid has area equal to its width times the average of its two heights.\nHalving the width of the pieces roughly cuts the error by four.\nFor $f(x) = x^2$ on $[0, 1]$ with two pieces, the trapezoids give $3/8$, slightly more than the true value $1/3$, because the parabola bends below its chords.\n",
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