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Lesson 10 · The grand connection · 8 min

The fundamental theorem

Derivatives measure change; integrals accumulate totals. The fundamental theorem of calculus reveals they are the same operation, run in opposite directions — and turns area problems into two function evaluations.

Build the intuition

Accumulation has a derivative — and it's the curve

Sweep x rightward, accumulating area A(x) under a curve f. How fast does A grow right now? By however tall f is right now — a tall curve pours in area quickly. In symbols: A′(x) = f(x). Accumulating, then differentiating, hands back the original. The loop closes.

ddx∫0xf(t) dt=f(x)\frac{d}{dx} \int_0^x f(t)\,dt = f(x)

The evaluation shortcut

Flip it around: to find the area under f from a to b, find any antiderivative F and compute F(b) − F(a). No rectangles, no slicing — two evaluations and a subtraction. Centuries of geometric struggle, collapsed into a move you can do in a margin.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Why this is reasonable

F′ = f means f is the rate at which F changes. Adding up all of F's instantaneous changes from a to b must give its total change: F(b) − F(a). The integral adds the changes; the difference is the total. Said aloud, it's almost obvious — that's the mark of a great theorem.

See it move

InteractiveThe fundamental theorem, watched live

A rate f(t) — area shading in

Total so far A(x) — the area, graphed

3
Accumulated area so far: 6.49. The bottom curve grows at rate 1.64 — the exact height of the top curve. Accumulating undoes rate-taking.

Sweep the shaded area and watch its running total graphed live below — growing exactly as fast as the top curve is tall.

A worked example

An area, in three lines

  1. Area under x² from 0 to 3:

    ∫03x2 dx\int_0^3 x^2\,dx

  2. Antiderivative: x³/3. Evaluate at both ends:

    333−033=9\frac{3^3}{3} - \frac{0^3}{3} = 9

  3. Exactly 9 — no approximation, no rectangles. The theorem did the infinite work for us.

The formula, earned

∫abf(x) dx=F(b)−F(a),F′=f\int_a^b f(x)\,dx = F(b) - F(a), \quad F' = f

Area = change in any antiderivative.

Out in the world

GPS distance

Your phone integrates velocity to track distance — but it doesn't add a billion rectangles. It leans on the fundamental theorem and antiderivative patterns to make the computation cheap enough for a chip in your pocket.

Common confusion, cleared

“Derivatives and integrals are separate subjects.”

They're inverse operations — like doing and undoing. The theorem is the official certificate of their marriage.

“The +C breaks F(b) − F(a).”

Any antiderivative works: the C in F(b) and the C in F(a) cancel in the subtraction. Pick the tidy one.

Recap

  • Differentiating an accumulation returns the original curve.
  • Areas via F(b) − F(a) — two evaluations, no slicing.
  • Derivative and integral are inverse operations: one subject.