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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $f \\colon C \\to \\mathbb{R}$ be [[continuous|0S9EK5MQPE]].\nFor a partition $a = t_0 \u003c t_1 \u003c \\dots \u003c t_k = b$ and points $\\tau_i \\in [t_{i-1}, t_i]$, let $\\Delta s_i$ be the length of the restriction of $\\gamma$ to $[t_{i-1}, t_i]$.\nThe line integral of $f$ along $\\gamma$ is the limit\n$$\\int_\\gamma f \\, ds = \\lim \\sum_{i=1}^{k} f(\\gamma(\\tau_i)) \\, \\Delta s_i,$$\ntaken as $\\max_i (t_i - t_{i-1}) \\to 0$.\n\n## Remarks\n\nThe limit exists, and it does not depend on the choice of the points $\\tau_i$.\nFor a $C^1$ curve, $\\int_\\gamma f \\, ds = \\int_a^b f(\\gamma(t)) \\, \\|\\gamma'(t)\\| \\, dt$. This is the [[parametric formula|0W3W1VYWBP]].\nFor $f = 1$ the line integral is the length of $\\gamma$. If $\\gamma$ is a wire with linear density $f$, the line integral is its mass.\nThe value does not change under a reparametrisation of $\\gamma$, in either direction.\n\n## Explanations\n\n### L1 human\n\nWalk along the curve, and at each small step multiply the length of the step by the value of $f$ there.\nThe sum of these products, in the limit of small steps, is the line integral with respect to arc length.\n",
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