In reply to Mark Janeba's question about how to convince students that the formula for the average of f(x) on the interval [a,b] is reasonable:
Actually, recently my Calc 2 students "convinced" me that the formula is reasonable. We were discussing a function that supposedly gave the temperature as a function of time over the course of a 24-hour day. When I wondered aloud what the evening weather reporters mean when they talk about the "average" temperature ... The students became intrigued by this and a lively discussion followed. They reasoned that to average something, we generally add everything up and divide by the total somehow. So they constructed a Riemann sum ... so much for "adding it all up". They argued for a while about what to divide by, and came to a consensus that it would be reasonable to divide by the length of the interval, or by |a-b|. The entire discussion took only about 30 minutes. It was one of those moments when I had the grace to stand out of the way and let the students develop the idea themselves.
BER (aka Barbara Reynolds, Cardinal Stritch College, Milwaukee, WI)