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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be continuous.\nThen there is $c \\in [a, b]$ with\n$$\\int_a^b f(x) \\, dx = f(c) \\, (b - a).$$\n\n## Remarks\n\nLet $m$ and $M$ be the minimum and the maximum of $f$, from the [[extreme value theorem|0WS1A8E2P9]]. By [[monotonicity|0MBWQKM8QR]], the average $\\frac{1}{b - a} \\int_a^b f$ lies in $[m, M]$. The [[intermediate value theorem|00KAFBR32E]] gives $c$.\nThe number $\\frac{1}{b - a} \\int_a^b f$ is the average value of $f$ on $[a, b]$.\nMathlib has a form for averages, `exists_eq_interval_average`.\n\n## Explanations\n\n### L1 human\n\nA continuous function takes its average value somewhere.\nThe area under the graph equals the area of a rectangle whose height is a value of $f$.\n",
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