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  "body": "\n## Statement\n\nLet $f \\colon [a, b] \\to \\mathbb{R}$ be Riemann integrable.\nThen $|f|$ is Riemann integrable on $[a, b]$, and\n$$\\Big| \\int_a^b f(x) \\, dx \\Big| \\le \\int_a^b |f(x)| \\, dx.$$\n\n## Remarks\n\nThe inequality follows from $-|f| \\le f \\le |f|$ and the [[monotonicity of the integral|0MBWQKM8QR]].\nThe converse of the first claim fails. On $[0, 1]$, let $f = 1$ on the rationals and $f = -1$ elsewhere. Then $|f| = 1$ is integrable, but $f$ is not.\nMathlib states the inequality as `intervalIntegral.norm_integral_le_integral_norm`.\n\n## Explanations\n\n### L1 human\n\nCancellation can only make an integral smaller in size.\nThe size of the integral is at most the integral of the size of the function.\n",
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