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  "body": "\n## Definition\n\nLet $a \u003c b$, let $f \\colon [a, b] \\to \\mathbb{R}$, and let $P = (x_0, \\dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.\nLet $t = (t_1, \\dots, t_n)$ be tags with $t_i \\in [x_{i-1}, x_i]$ for each $i$.\nThe Riemann sum of $f$ for the tagged partition $(P, t)$ is\n$$S(f, P, t) = \\sum_{i=1}^{n} f(t_i) (x_i - x_{i-1}).$$\n\n## Remarks\n\nThe left rule takes $t_i = x_{i-1}$, the right rule takes $t_i = x_i$, and the midpoint rule takes the midpoints.\nFor bounded $f$, each Riemann sum lies between the lower and the upper [[Darboux sum|0QRQVH0CRW]] for $P$.\nMathlib has a general form for boxes, `BoxIntegral.integralSum`.\n\n## Explanations\n\n### L1 human\n\nA Riemann sum adds up areas of rectangles.\nEach rectangle stands on a piece of the partition, and its height is the value of $f$ at the chosen tag.\n",
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