I was downloading something yesterday and noticed that my downloading program tells me the download speed, and it updates it every half second or so. It also tells me how much of the file I’ve downloaded, total (e.g. 29.6 MB out of 64 MB, 46.3% completed).
This is a perfect example of integration as accumulation! The integral gives you the total amount of the file the program has downloaded. The graph above — created by a bittorrent program called 
This actually would be a great way to have students come up with the conceptual idea of Riemann sums themselves: given a thousand data points, collected every half-second, of the download speed, what would be a good way to figure out how much has been downloaded. How would you estimate it?
And you could extend it to say: if you wanted a quicker but less accurate way to come up with how much has been downloaded, how would you do that? Students might say, “take every fifth data point” or “average every five data points” or come up with some other interesting method!
Or you could ask if there is a more accurate way to come up with how much has been downloaded. And there’s a good chance, with some requisite prompting and asking the right questions, that they could come up with the Trapezoidal Rule. And then you could segue into Simpson’s Rule.
Note: I know, I know, you could do the same thing with a speed v. time graph (giving you distance), or any other number of graphs. But I like this. It comes naturally out of things we do everyday!
[1] I cribbed this from here.
Continuous Everywhere but Differentiable Nowhere 
