Well, no. Given your number of customers as a function of price D(p), you aren't necessarily trying to maximize p D(p). It's not completely obvious, but in principle you can get ∫ D(p) dp out of those people, if everyone paid what they were willing to pay.
In order to pursue this goal it is very common to make a finer approximation to ∫ D(p) dp by creating several price-points. You see this all the time on Kickstarter for example, "donate some extra and we'll send you a copy of the source code." And one of these ways is "donate some extra and you'll get to see the finished product first," which is one of the solutions given above: charge $15 when it first comes out, then reduce to $5 over time.
There's also a decay function with a roughly exponential envelope that covers the number of people possibly interested as a function of time.
For a documentary that pretty much only appeals to science and engineering geeks, hitting the front page of HN is the peak interest at t=0. Now what. Well it starts decaying. Might go up a bit, but in a week that tail is going to be dominant and the potential sales a minute fraction of those possible TODAY if a sales conversion occurs.
Most films are highly perishable. They don't make a lot after initial interest and excitement wanes as new films are always coming out.
Right now I can buy a couple DVDs I want with shipping for the same $15 of a download of a documentary that people are saying is sort of OK but has slow moments, or I can buy a month's Netflix and realize the real limit is I only have so much time to watch things.
Maybe. It's worth pointing out that the PhD Movie was actually released many months ago, has already screened in many college campuses, and is now receiving a buzz of revived interest from us today. So this isn't t = 0 but t = 1, in the appropriate units.
The DVDs you propose buying are actually probably one of the most successful implementations of this strategy, and the key fact is that sometime a year down the line you say to yourself, "hey, I never got around to seeing that movie, it was too expensive, or they only had it in 3D and 3D makes my head hurt, or any number of other things -- why don't I just buy it at the reduced price on DVD?" It works.
(I'd say it works "very well" but honestly, DVD sales are pretty thinly reported across the industry, so nobody really knows how well it works in terms of hard numbers.)
That's the theory. In practice, people who think that $15 are too much, will either forget about it (and not come back when its cheaper), or they will pirate it.
(1) As for forgetting, well, good companies often market their price reductions. I mean, famously, Apple released two iPhones in 2007, a 4GB one at $500 and an 8GB one at $600, and rapidly killed the first and reduced the second to $400, after about two months. It made a terrific marketing splash, lots of people bought the $400 one. Granted, this sort of marketing is easy for them because they're Apple and everybody is always listening for news on them, but my point is just that "we're reducing our price!" can be a subject worthy of a new press release and a new Hacker News bump.
You may also remember that so many people criticized them for the above, that Apple gave away $100 certificates to the people who bought at the $600 price point. There is a valid criticism of what I have said above which nobody has yet mentioned: which is that people like to get a "good deal" and think of you better as a company if you provide that to them. As you try to extract out ∫ D(p) dp from people, you are also squeezing a commensurate amount of happiness out of them, and that can be bad for long-term customers.
(2) The top isoHunt torrent lists 20-ish torrenters, so I'm not sure whether piracy is a valid concern on this matter.
In order to pursue this goal it is very common to make a finer approximation to ∫ D(p) dp by creating several price-points. You see this all the time on Kickstarter for example, "donate some extra and we'll send you a copy of the source code." And one of these ways is "donate some extra and you'll get to see the finished product first," which is one of the solutions given above: charge $15 when it first comes out, then reduce to $5 over time.