  We provide an elementary geometric derivation of the Kac integral formula for
the expected number of real zeros of a random polynomial with independent
standard normally distributed coefficients. We show that the expected number of
real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$
projected onto the surface of the unit sphere, divided by $\pi$. The
probability density of the real zeros is proportional to how fast this curve is
traced out. We then relax Kac's assumptions by considering a variety of random
sums, series, and distributions, and we also illustrate such ideas as integral
geometry and the Fubini-Study metric.
