Motivation of the Gaussian Integral I read on Wikipedia that Laplace was the first to evaluate
$$\int\nolimits_{-\infty}^\infty e^{-x^2} \, \mathrm dx$$
Does anybody know what he was doing that lead him to that integral? Even better, can someone pose a natural problem that would lead to this integral?
Edit: Many of the answers make a connection to the normal distribution, but then the question now becomes: Where does the density function of the normal distribution come from? Mike Spivey's answer is in the spirit of what I am looking for: an explanation that a calculus student might understand.

The function $e^{-x^2}$ is natural for investigation for lots of different reasons. One reason is that, depending on your normalization, it is essentially a fixed point of the Fourier Transform.  That is,
$$
\int_{\mathbb{R}^n} e^{-\pi x^2} e^{-2\pi ix\cdot t}\mathrm{d}x=e^{-\pi t^2}
$$
Another reason is tied to the Central Limit Theorem. Suppose that $f$ satisifies $\int_{\mathbb{R}^n}f(x)\;\mathrm{d}x=1$, $\int_{\mathbb{R}^n}x\;f(x)\;\mathrm{d}x=0$, and $\int_{\mathbb{R}^n}|x|^2\;f(x)\;\mathrm{d}x=1$ (these can be attained by translating and scaling the domain and scaling the range of $f$). Let $f^{\;*k}$ be the convolution of $f$ with itself $k$ times.  Then $k^{n/2}f^{\;*k}(x\sqrt{k})\to \frac{1}{\sqrt{2\pi}^n}e^{-x^2/2}$ as $k\to\infty$.
