The Fundamental Theorem of Calculus

Differentiation found slopes. Integration found areas. They looked like separate subjects.

The first part

If A(x) is the area accumulated up to x, then A′(x) = f(x). Accumulating and differentiating undo each other.

The second part

The definite integral from a to b equals F(b) − F(a), where F is any antiderivative of f.

Why this is the centre of the subject

An infinite sum of infinitely thin rectangles is computed by subtracting two numbers.

The two halves of the subject are one

Differentiation and integration were developed as separate problems — tangents and areas. The fundamental theorem says each undoes the other. That connection is the central result of the whole subject.

The first part

If you define a function as the accumulated area under f up to x, its derivative is f. Accumulating and then differentiating returns you to where you started, which is far from obvious in advance.

The second part

The definite integral from a to b equals F(b) − F(a) for any antiderivative F. This converts an infinite limiting process into two evaluations and a subtraction, which is why it transformed mathematics.

Why it mattered so much

Before it, areas were computed by exhaustion — laborious limits done case by case. After it, area became a search for an antiderivative. Problems that had taken Archimedes pages became a line of algebra.