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  "body": "\n## Statement\n\nLet $R = [a, b] \\times [c, d]$, and let $f \\colon R \\to \\mathbb{R}$ be Riemann integrable on $R$.\nSuppose that $y \\mapsto f(x, y)$ is integrable on $[c, d]$ for each $x \\in [a, b]$, and that $x \\mapsto f(x, y)$ is integrable on $[a, b]$ for each $y \\in [c, d]$.\nThen\n$$\\iint_R f \\, dA = \\int_a^b \\int_c^d f(x, y) \\, dy \\, dx = \\int_c^d \\int_a^b f(x, y) \\, dx \\, dy.$$\nIn particular, this holds for each [[continuous|0S9EK5MQPE]] function $f$ on $R$.\n\n## Remarks\n\nThe left side is the [[double integral|0NKBMFC18R]], and the other two are [[iterated integrals|0P4WX7Y502]].\nThe Lebesgue version: let $f$ be Lebesgue measurable on $\\mathbb{R}^n \\times \\mathbb{R}^m$ with $\\int |f| \u003c \\infty$. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable.\nWithout integrability of $|f|$, the [[iterated integrals can differ|0GJZE2E4RX]]. For non-negative functions, [[Tonelli's theorem|067SYD2J4B]] removes the integrability hypothesis.\nEuler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.\n\n## Explanations\n\n### L1 human\n\nTo integrate over a rectangle, you can slice it into lines, integrate along each line, and then integrate the results.\nFubini's theorem says that both slicing directions give the same answer as the double integral, for well-behaved functions.\n",
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