Riemann Sum
We have found that we can look at accumulation problems (if we know how much is being added or subtracted each moment, how much stuff is there?) as area under a graph.
But the area under most graphs is a difficult problem. Newton and Leibniz figured out that they could approximate with rectangles and then imagine the approximation being perfect by having an "infinite number" of rectangles. Later (Georg Friedrich) Bernhard Riemann made this hand wavy idea precise.
in this applet you can choose the function and the number of rectangles. Some things to look at include the left/middle/right control and how it affects the estimate, how accurate the estimates are for the number of rectangles. And always, what do you notice?