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  "body": "\n## Definition\n\nLet $a \u003c b$, let $f \\colon [a, b] \\to \\mathbb{R}$ be bounded, and let $P = (x_0, \\dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.\nFor each $i$, let $M_i = \\sup f([x_{i-1}, x_i])$ and $m_i = \\inf f([x_{i-1}, x_i])$.\nThe upper and lower Darboux sums of $f$ for $P$ are\n$$U(f, P) = \\sum_{i=1}^{n} M_i (x_i - x_{i-1}), \\qquad L(f, P) = \\sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$\n\n## Remarks\n\nAlways $(b - a) \\inf f \\le L(f, P) \\le U(f, P) \\le (b - a) \\sup f$.\nFor a [[refinement|0EEKF7YFBW]] $Q$ of $P$, we have $L(f, P) \\le L(f, Q)$ and $U(f, Q) \\le U(f, P)$.\nSo $L(f, P) \\le U(f, Q)$ for any two partitions $P$ and $Q$.\n\n## Explanations\n\n### L1 human\n\nThe upper sum covers the graph with rectangles that reach its highest point on each piece.\nThe lower sum uses rectangles that reach its lowest point on each piece.\n",
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