{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0GTZ2F3PQY"
    },
    {
      "field": "uses",
      "target": "0GVMR0C3S6"
    },
    {
      "field": "uses",
      "target": "0JX86PSSA9"
    },
    {
      "field": "uses",
      "target": "0MPZGAZ2GK"
    },
    {
      "field": "uses",
      "target": "0VEKNG02W0"
    }
  ],
  "body": "\n## Definition\n\nLet $B = [a_1, b_1] \\times [a_2, b_2] \\times [a_3, b_3]$ with $a_i \u003c b_i$, and let $f \\colon B \\to \\mathbb{R}$ be bounded.\nPartition each interval $[a_i, b_i]$, and let $B_1, \\dots, B_N$ be the resulting closed sub-boxes, with volumes $\\Delta V_1, \\dots, \\Delta V_N$.\nFor points $p_k \\in B_k$, the sum $S = \\sum_{k=1}^{N} f(p_k) \\, \\Delta V_k$ is a Riemann sum of $f$.\nThe function $f$ is Riemann integrable on $B$ if there is $I \\in \\mathbb{R}$ with this property. For each $\\varepsilon \u003e 0$, there is $\\delta \u003e 0$ with $|S - I| \u003c \\varepsilon$ for all Riemann sums whose sub-boxes have diameter less than $\\delta$.\nThen $I$ is the triple integral of $f$ over $B$, written $\\iiint_B f(x, y, z) \\, dV$.\n\n## Remarks\n\nThis is the case $n = 3$ of the multiple integral over a box. The case $n = 2$ is the [[double integral|0NKBMFC18R]].\nFor a bounded solid $E \\subseteq B$, the integral over $E$ is the integral over $B$ of $f$ times the indicator function of $E$. The volume of $E$ is $\\iiint_E 1 \\, dV$, when this integral exists.\nBy [[Fubini's theorem|0R0T1VWH5X]], for continuous $f$ the triple integral equals an iterated integral. For example, $\\iiint_{[0,1]^3} x y z \\, dV = (1/2)^3 = 1/8$.\n\n## Explanations\n\n### L1 human\n\nA triple integral adds up the values of $f$ over a solid, each weighted by the volume of a small piece.\nIf $f$ is a density, the triple integral is the total mass of the solid.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0698M7C7AX",
    "kind": "definition",
    "msc": [
      "26B15"
    ],
    "name": "Triple integral",
    "sources": [
      {
        "locator": "Mathematical definition",
        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
      },
      {
        "locator": "Triple integral",
        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
      }
    ],
    "tag": "0GTBGMZXV8",
    "uses": [
      "0NKBMFC18R",
      "0JPCHTXWG0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0GTBGMZXV8/",
  "links": [
    {
      "field": "defines",
      "target": "0698M7C7AX"
    },
    {
      "field": "uses",
      "target": "0NKBMFC18R"
    },
    {
      "field": "uses",
      "target": "0JPCHTXWG0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0GTBGMZXV8.md",
  "msc": [
    "26B15"
  ],
  "name": "Triple integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0GT/0GTBGMZXV8.md",
  "status": null,
  "tag": "0GTBGMZXV8",
  "uri": "m2://0GTBGMZXV8",
  "url": "https://math2.tamnd.com/wiki/triple-integral-definition/",
  "writing": {
    "tag": "0GTBGMZXV8",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}