The reason I’m wondering about 1619 is that the events of 1619 were in large part determined by events outside the U.S., so pinning the responsibility on the U.S. at that point seems…odd. I certainly get 1776 — at that point, the U.S. officially took ownership, so it was certainly on them from that point on (and effectively, probably, for some time before). But anyway, that’s a secondary point.

Let’s deal with the primary metric first.

Suppose for the sake of argument, that there had been only one black guy enslaved (the only black guy in what would become the U.S.) and whose life was basically considered worthless in 1619, and he lived until, say, 1689; and when he died, they got another; and so on and so forth until 1989, and at that point 35 million black people came into the country whose lives were each valued twice as high as white lives, and it’s been that way for the past 32 years. Your calculation seems like it would say that we are still running a massive deficit, despite the historical reality being that there were 360 person-years of devaluation experienced (1 person at a time x 360 years) vs. 1.12 billion person-years of double-valuation experienced (35 million people at a time x 32 years).

Somehow that seems intuitively wrong, wouldn’t you say?

Also, suppose that we have two graphs: one oscillates wildly from “black lives are worthless” (value = 0x white life value) to “black lives are doubly precious” (value = 2x white life value), but it does so sinusoidally (f(t) = 1 + sin(2*pi*t/T) for some period T) so that the integral is the same as the constant graph (value of black life = value of white life, f(t) = 1). Your integral would say these two things are equally desirable.

Somehow that also seems intuitively wrong. No?

Now, neither of these hypotheticals are what actually happened. But if the analytic approach doesn’t work right on these admittedly somewhat weird cases, we should probably understand in detail what went wrong before we trust it to give helpful results for the actual situation.

(I also don’t think it’s just futzing around with math. Math reflects our intuitions, if used carefully, and can help get the scale of things right when the scale is hard to grasp with one’s intuition.)