I'm liking how I'm introducing integrals in calculus. I figured we'd concentrate on 1 to 2 functions and work them in all sorts of ways to build up to an understanding of what's so cool about the (first?) fundamental theorem of calculus. So we approximated area, we brainstormed ways to make it more accurate, we found lower and upper bounds to exact area under the curve, and I then built up how to use limits and summation to get the exact area.
Sheesh, these things always take longer than you think they will. In my mind (hah!) I figured they could finish up simplifying this one equation in class and then I'd assign the 2nd problem for homework. Hah. Hah. What took most of them all period was to simplify an equation of this form (note that it's not a matter of calculus right now, it's basic algebra .... or so ... FOILing, distributing, adding fractions, simplifying):
[1/2 (3i/n - 2)(3i/n - 2) + 1](3/n).
Okay, ick (sort of), but doable ..... (apparently doable in 40 minutes and not the 5 or so minutes I envisioned in my fantasy teaching mind).