{
  "aliases": [
    "integral-over-a-normal-domain",
    "type-i-and-type-ii-regions"
  ],
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  "body": "\n## Summary\n\nTo integrate a function $f$ over a bounded region $D$ that is not a rectangle, take a rectangle $R \\supseteq D$. Then integrate over $R$ the function equal to $f$ on $D$ and to $0$ on $R \\setminus D$.\nFor a region between two graphs, $D = \\{(x, y) : a \\le x \\le b, \\ \\varphi_1(x) \\le y \\le \\varphi_2(x)\\}$ with $\\varphi_1 \\le \\varphi_2$ continuous, a continuous $f$ gives $\\iint_D f \\, dA = \\int_a^b \\int_{\\varphi_1(x)}^{\\varphi_2(x)} f(x, y) \\, dy \\, dx$.\n\n## Remarks\n\nSuch a region is normal with respect to the $x$-axis. A region of the form $\\{c \\le y \\le d, \\ \\psi_1(y) \\le x \\le \\psi_2(y)\\}$ is normal with respect to the $y$-axis.\nThe definition does not depend on the choice of $R$. For continuous $f$, the integral exists when the boundary of $D$ has [[Jordan content|0KTGJW1ZTH]] zero.\nExample: for $D = \\{0 \\le x \\le 1, \\ x^2 \\le y \\le 1\\}$, $\\iint_D x \\, dA = \\int_0^1 x (1 - x^2) \\, dx = 1/4$.\nThe [[double integral|0WAR4F23B4]] over $D$ is computed as an [[iterated integral|0WZCHE1ZF9]] with variable bounds.\n\n## Explanations\n\n### L1 human\n\nMost regions in practice are not rectangles, but many lie between two curves.\nThen the inner integral runs from the lower curve to the upper curve, and the outer integral runs over the range of the other variable.\n",
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        "locator": "Double integral over a normal domain",
        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
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  "name": "Integration over a non-rectangular region",
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