Notes

The Fundamental Theorem of Calculus

last edited Mathematics

The two operations of calculus are inverse to one another. Making that precise is the content of the fundamental theorem, and the statement is worth reading carefully1.

The result connects two definitions that look nothing alike: the derivative as a limit of difference quotients, and the integral as a limit of Riemann sums.

The statement

Theorem 1.1 (Fundamental Theorem of Calculus)

Let 𝑓 be continuous on [𝑎,𝑏] and define 𝐹(𝑥) =∫𝑥𝑎𝑓(𝑡) 𝑑𝑡. Then 𝐹 is differentiable on (𝑎,𝑏) and 𝐹′(𝑥) =𝑓(𝑥).

Proof

Fix 𝑥 ∈(𝑎,𝑏) and let ℎ ≠0 be small enough that 𝑥 +ℎ ∈(𝑎,𝑏). Then

𝐹(𝑥+ℎ)−𝐹(𝑥)ℎ=1ℎ∫𝑥+ℎ𝑥𝑓(𝑡)𝑑𝑡.(1)

By continuity of 𝑓 at 𝑥, the right-hand side of (1) tends to 𝑓(𝑥) as ℎ →0.

A consequence

Corollary 2.1

If 𝐺 is any antiderivative of 𝑓 then ∫𝑏𝑎𝑓 =𝐺(𝑏) −𝐺(𝑎).

This is what Theorem 1.1 is for in practice: it turns the evaluation of an integral into the search for an antiderivative.

Notes

  1. The continuity hypothesis matters. Dropping it gives a version that holds almost everywhere, which is a genuinely different theorem. ↑