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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{R}^2$ be a region that has two descriptions,\n$$D = \\{(x, y) : a \\le x \\le b, \\ \\varphi_1(x) \\le y \\le \\varphi_2(x)\\} = \\{(x, y) : c \\le y \\le d, \\ \\psi_1(y) \\le x \\le \\psi_2(y)\\},$$\nwhere $\\varphi_1 \\le \\varphi_2$ are continuous on $[a, b]$ and $\\psi_1 \\le \\psi_2$ are continuous on $[c, d]$.\nLet $f \\colon D \\to \\mathbb{R}$ be continuous.\nThen\n$$\\int_a^b \\int_{\\varphi_1(x)}^{\\varphi_2(x)} f(x, y) \\, dy \\, dx = \\int_c^d \\int_{\\psi_1(y)}^{\\psi_2(y)} f(x, y) \\, dx \\, dy,$$\nand both equal the double integral of $f$ over $D$.\n\n## Remarks\n\nThe proof applies [[Fubini's theorem|0R0T1VWH5X]] on a rectangle that contains $D$ to $f$ times the indicator function of $D$.\nThe main work is to describe $D$ again in the other order. One way is to sketch the region.\nExample: $\\int_0^1 \\int_x^1 e^{y^2} \\, dy \\, dx$ has no elementary inner antiderivative. The region is $0 \\le x \\le y \\le 1$, so the integral equals $\\int_0^1 \\int_0^y e^{y^2} \\, dx \\, dy = \\int_0^1 y e^{y^2} \\, dy = (e - 1)/2$.\nFor unbounded regions or discontinuous integrands, a change of order needs extra hypotheses.\n\n## Explanations\n\n### L1 human\n\nAn iterated integral over a region can sometimes be hard in one order and easy in the other.\nChanging the order means describing the same region by slicing it in the other direction, and then integrating in the new order.\n",
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