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  "body": "\n## Definition\n\nLet $R = [a, b] \\times [c, d]$ and let $f \\colon R \\to \\mathbb{R}$.\nSuppose that, for each $x \\in [a, b]$, the function $y \\mapsto f(x, y)$ is integrable on $[c, d]$, and that $F(x) = \\int_c^d f(x, y) \\, dy$ is integrable on $[a, b]$.\nThen the iterated integral of $f$, first in $y$ and then in $x$, is\n$$\\int_a^b \\int_c^d f(x, y) \\, dy \\, dx = \\int_a^b \\left( \\int_c^d f(x, y) \\, dy \\right) dx = \\int_a^b F(x) \\, dx.$$\nThe iterated integral in the other order, $\\int_c^d \\int_a^b f(x, y) \\, dx \\, dy$, is defined in the same way.\n\n## Remarks\n\nIterated integrals in three or more variables are defined by repeating this step.\nThe inner bounds may depend on the outer variable, as in $\\int_0^1 \\int_0^x f(x, y) \\, dy \\, dx$. This integrates over the triangle $0 \\le y \\le x \\le 1$.\nAn iterated integral is different in principle from the [[double integral|0NKBMFC18R]], but [[Fubini's theorem|0R0T1VWH5X]] says that they are equal under suitable hypotheses.\nThe two orders of integration can give [[different values|0GJZE2E4RX]] when the hypotheses fail.\n\n## Explanations\n\n### L1 human\n\nTo compute an iterated integral, work from the inside out.\nFirst integrate in $y$ as if $x$ were a constant; the result is a function of $x$ only. Then integrate this function in $x$.\n",
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