Simpson and Improperness

The most beautiful aspect of math is that the more about it you learn, the rate at which it becomes more interesting grows exponentially (that is quite nearly a pun); and, I've just scratched the surface. I was reading about Simpson's rule the other day, and it's pretty cool. You can approximate the area under the curve using a simple series, which is derived from fitting parabola approximations to the curve. What's even more amazing is that approximations from Simpson's rule just happen to be the weighted average of those from the midpoint and trapezoidal rules (Sn = (2/3)Mn + (1/3)Tn). ALWAYS. I haven't a clue why, but it's pretty amazing.

Even more amazing are improper integrals. Essentially, if you have a function with an asymptote, the area under the graph from a fixed point to the asymptote (among other cases) sometimes is a finite number. So, even though the line approaches infinity, the area under it can be finite (if the function is convergent). It's so bizarre.

Many of you are probably already aware of both of these principlesand think I'm silly, but the coincidence and counterintuitiveness (?) of math is constantly fascinating to me.