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  "body": "\n## Summary\n\nLaplace's method approximates integrals of the form $\\int_a^b e^{M f(x)}\\, dx$ when the parameter $M$ is large.\nFor large $M$ almost all of the integral comes from a small neighbourhood of the point where $f$ is largest.\nThere $f$ is close to a downward parabola.\nReplacing $f$ by this parabola turns the integral into a Gaussian integral, which gives the leading term of an [[asymptotic equivalence|00SB5EA3VW]].\n[[Stirling's formula|0KC0F51T6A]] is the classical application.\n\n## Explanations\n\n### L1 human\n\nRaise $e$ to the power $M f(x)$ with a huge $M$: the values of $x$ where $f$ is largest become far more important than all others.\nThe graph of $e^{M f(x)}$ turns into a tall, thin bump above the maximum point.\nNear its top the bump looks like a bell curve, whose area is known exactly.\nSo the integral is about the height of the bump times a width proportional to $1/\\sqrt{M}$.\n",
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