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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be continuous.\nThe arc length of $\\gamma$ is\n$$L(\\gamma) = \\sup_{P} \\sum_{i=1}^{k} \\| \\gamma(t_i) - \\gamma(t_{i-1}) \\|,$$\nwhere the supremum runs over all [[partitions|069P20KYQT]] $P = (t_0, \\dots, t_k)$ of $[a, b]$.\nThe curve $\\gamma$ is rectifiable if $L(\\gamma) \u003c \\infty$.\n\n## Remarks\n\nEach sum is the length of a polygonal path through points of the curve. Adding points does not decrease this length.\nIf $\\gamma$ is continuously differentiable, then $L(\\gamma) = \\int_a^b \\| \\gamma'(t) \\| \\, dt$.\nFor the graph of a $C^1$ function $f$ on $[a, b]$, this gives $L = \\int_a^b \\sqrt{1 + f'(x)^2} \\, dx$.\nSome continuous curves are not rectifiable, for example the graph of the [[Weierstrass function|01A0MK1730]].\n\n## Explanations\n\n### L1 human\n\nThe arc length of a curve is the length of a string laid along it.\nWe approximate the curve by straight segments and take the best approximation.\n",
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