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  "body": "\n## Summary\n\nThe Fresnel integrals $S(x) = \\int_0^x \\sin(t^2)\\, dt$ and $C(x) = \\int_0^x \\cos(t^2)\\, dt$ are integrals of [[sine and cosine|08WFZP6P89]] of a square.\nThey arise in the optics of diffraction near an edge, studied by Augustin-Jean Fresnel.\nThe plane curve $(C(t), S(t))$ is the Euler spiral, whose curvature grows linearly with arc length.\nBoth integrals tend to $\\sqrt{\\pi/8}$ as $x \\to \\infty$.\n\n## Explanations\n\n### L1 human\n\nThe function $\\sin(t^2)$ oscillates faster and faster as $t$ grows.\nIts area from $0$ to $x$ therefore wobbles less and less and settles down to a limit.\nIf one draws the point $(C(t), S(t))$ as $t$ increases, one gets a spiral that winds tighter and tighter around a point.\nRoad and railway designers use this spiral to pass smoothly from a straight line into a curve.\n",
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