{
  "aliases": [
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  "body": "\n## Summary\n\nAn improper multiple integral extends the Riemann integral to unbounded regions or unbounded functions. It is a limit of integrals over bounded regions where the function is bounded.\nFor a non-negative function $f$ on $\\mathbb{R}^n$ that is Riemann integrable on each cube $[-a, a]^n$, the improper integral is $\\int_{\\mathbb{R}^n} f = \\lim_{a \\to \\infty} \\int_{[-a, a]^n} f$, when this limit exists.\n\n## Remarks\n\nFor non-negative functions, the limit does not depend on the choice of the exhausting sequence of bounded sets. For functions of both signs one usually requires that $\\int |f|$ is finite.\nExample: in polar coordinates, $\\iint_{\\mathbb{R}^2} e^{-(x^2 + y^2)} \\, dA = \\lim_{a \\to \\infty} \\int_0^{2\\pi} \\int_0^a e^{-r^2} r \\, dr \\, d\\theta = \\pi$.\nExample with a singularity: $\\iint_{\\|x\\| \\le 1} \\|x\\|^{-s} \\, dA$ is finite for $s \u003c 2$ and infinite for $s \\ge 2$, by the same polar computation.\nAn improper [[double integral|0WAR4F23B4]] can fail to exist even when both iterated integrals exist, if $\\int |f| = \\infty$.\n\n## Explanations\n\n### L1 human\n\nThe Riemann integral needs a bounded function on a bounded region.\nYou may want to integrate over all of the plane, or near a point where $f$ blows up. Then integrate over larger and larger bounded pieces and take the limit.\n",
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    "name": "Improper multiple integral",
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      {
        "locator": "Multivariable improper integrals",
        "url": "https://en.wikipedia.org/wiki/Improper_integral"
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        "locator": "Improper integrals over arbitrary domains",
        "url": "https://en.wikipedia.org/wiki/Improper_integral"
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        "locator": "Multiple improper integral",
        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
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  "msc": [
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