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  "body": "\n## Definition\n\nLet $E \\subseteq \\mathbb{R}^n$ be a bounded Jordan measurable set, and let $\\delta \\colon E \\to [0, \\infty)$ be a continuous density with total mass $M = \\int_E \\delta(x) \\, dx \u003e 0$.\nThe centre of mass of $E$ with density $\\delta$ is the point $\\bar{x} = (\\bar{x}_1, \\dots, \\bar{x}_n)$ with\n$$\\bar{x}_i = \\frac{1}{M} \\int_E x_i \\, \\delta(x) \\, dx, \\qquad i = 1, \\dots, n.$$\nIf $\\delta$ is constant, $\\bar{x}$ is the centroid of $E$, and it does not depend on the value of the constant.\n\n## Remarks\n\nFor $n = 3$ the integrals are [[triple integrals|0GTBGMZXV8]], and for a plane lamina, $n = 2$, they are [[double integrals|0NKBMFC18R]].\nFor point masses $m_1, \\dots, m_k$ at $p_1, \\dots, p_k$, the centre of mass is $\\sum_j m_j p_j / \\sum_j m_j$. The definition above replaces the sum by an integral.\nExample: the half disc $\\{x^2 + y^2 \\le a^2, \\ y \\ge 0\\}$ with constant density has centroid $(0, 4a / (3\\pi))$.\nIf $E$ and $\\delta$ are symmetric under a reflection, then $\\bar{x}$ lies in the mirror of that reflection.\n\n## Explanations\n\n### L1 human\n\nThe centre of mass is the balance point of a body.\nIf the body is supported at that point, gravity does not turn it in any direction.\n",
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