1. The function P(x) = e-² is fundamental in probability. (a) Sketch the graph of P(x). Explain why it is called a "bell curve." (b) Compute I = Le e-2² de using the following brilliant strategy of Gauss: i. Instead of computing I, compute 1² = ( ²² dx) (e-² dy). -z² ii. Rewrite 1² as an integral of the form f(x,y) dA where R is the entire Cartesian plane. iii. Convert that integral to polar coordinates. iv. Evaluate to find 1². Deduce the value of I. Amazingly, it can be mathematically proven that there is NO elementary function Q(x) (that is, function built up from sines, cosines, exponentials, and roots using "usual" operations) for which Q'(x) = P(x).

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