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  "body": "\n## Statement\n\nLet $0 \\le r_1 \u003c r_2$ and $\\alpha \u003c \\beta \\le \\alpha + 2\\pi$, and let $D = \\{(r \\cos \\theta, r \\sin \\theta) : r_1 \\le r \\le r_2, \\ \\alpha \\le \\theta \\le \\beta\\}$.\nLet $f \\colon D \\to \\mathbb{R}$ be continuous.\nThen\n$$\\iint_D f(x, y) \\, dx \\, dy = \\int_\\alpha^\\beta \\int_{r_1}^{r_2} f(r \\cos \\theta, r \\sin \\theta) \\, r \\, dr \\, d\\theta.$$\n\n## Remarks\n\nThis is the [[change of variables formula|00N5FHP1CV]] for the [[polar coordinate map|0RJZV05MJA]], whose Jacobian determinant is $r$. The boundary rays and the origin have area zero and do not change the integral.\nExample: the area of the disc of radius $a$ is $\\int_0^{2\\pi} \\int_0^a r \\, dr \\, d\\theta = \\pi a^2$.\nFor a region $\\alpha \\le \\theta \\le \\beta$, $0 \\le r \\le g(\\theta)$ with $g$ continuous, the formula gives the area $\\tfrac{1}{2} \\int_\\alpha^\\beta g(\\theta)^2 \\, d\\theta$.\nThe Gaussian integral follows from $\\iint_{\\mathbb{R}^2} e^{-(x^2 + y^2)} \\, dx \\, dy = \\int_0^{2\\pi} \\int_0^\\infty e^{-r^2} r \\, dr \\, d\\theta = \\pi$. Hence $\\int_{-\\infty}^{\\infty} e^{-x^2} \\, dx = \\sqrt{\\pi}$.\n\n## Explanations\n\n### L1 human\n\nIn polar coordinates a small piece of the plane is almost a rectangle with sides $dr$ and $r \\, d\\theta$.\nSo its area is $r \\, dr \\, d\\theta$, and the factor $r$ appears in the integral.\n",
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  "msc": [
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  "name": "Integration in polar coordinates",
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