20  Integration

Substitution, integration by parts, partial fractions — the techniques aren’t this chapter’s subject; you already have them. What integration structurally is, and why it turns out to be the inverse of Differentiation — a fact that is not remotely obvious from the two problems’ descriptions — is.

20.1 A completely different-looking problem

Differentiation asks about instantaneous rate of change. Integration, on its face, asks something unrelated: how much area sits under a curve? Slice the interval \([a,b]\) into \(n\) pieces, build a rectangle over each piece using the function’s height there, sum the rectangles’ areas, and take the limit as the slices get arbitrarily fine. That’s a Riemann sum, and the integral

\[\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i)\,\Delta x\]

is defined as its limit — another limit, in exactly Limits and Continuity’s sense, this time of a sequence of increasingly fine approximating sums rather than of a difference quotient.

Not every bounded function has this limit exist — a function wild enough (discontinuous on a large enough set of points) can fail to be Riemann integrable at all. Fixing exactly which functions are integrable, by building the integral on measure theory instead of on Riemann sums, is Henri Lebesgue’s contribution (1902) — previewed in Advanced Topics; the Riemann integral above is what this chapter develops, and it’s sufficient for everything the rest of this site currently needs.

20.2 The theorem that unifies the two problems

The fundamental theorem of calculus says that if \(F\) is any antiderivative of \(f\) (meaning \(F' = f\)), then

\[\int_a^b f(x)\,dx = F(b) - F(a).\]

Sit with how strange this is before it becomes routine: one construction (area, built from sums of rectangles) and another (rate of change, built from difference quotients) turn out to be exact inverses of each other. Nothing in either definition — Riemann sums on one side, difference quotients on the other — makes that obviously true. This theorem is what makes “calculus” one subject rather than two, and establishing it, independently, is the achievement usually credited jointly to Newton and Leibniz in the 1660s–70s (and the source of a famously bitter priority dispute between them and their respective camps) — another two centuries before Riemann’s 1850s definition of the integral this chapter actually used to state the theorem rigorously. The historical-versus-formal gap the map keeps returning to shows up here in an unusually direct form: the result is 17th-century; the definition it’s stated with is 19th-century.

20.3 Why the standard techniques work, restated

Every integration technique you already know is the fundamental theorem read backwards. Substitution is the chain rule, reversed. Integration by parts is the product rule, reversed. This isn’t a new set of facts to learn — it’s the same rules from Differentiation, run in the other direction, which is exactly what “integration undoes differentiation” predicts they’d have to be.

20.4 What follows, briefly

Improper integrals extend the definition to infinite intervals or unbounded functions, as a limit of ordinary integrals over growing finite regions — another instance of the same “define the hard case as a limit of easier cases” move this entire chapter has been making. The deeper fix for which functions are integrable at all — Lebesgue integration, built on measure theory — is previewed in Advanced Topics rather than developed here.

20.5 If you’ve computed integrals

You already know \(\int_a^b f(x)\,dx\) as “area under the curve” and as \(F(b)-F(a)\). What you may not have had explicit: the first is the actual definition, built from Riemann sums exactly the way Limits and Continuity builds any limit; the second is a theorem about that definition, not a restatement of it — the fundamental theorem of calculus is the genuinely non-obvious claim connecting the two.

20.6 Where this leads

Differential Equations, next, is where integration reappears as a tool: an equation relating a function to its own derivative, solved — when it can be solved explicitly at all — by integrating your way back from the derivative to the function.