The Fundamental Theorem, Explained

FTC Part 1: F(t) = ∫₀ᵗ f(x) dx — move t to see the accumulated area

f(x) = x² F(x) = x³/3

t = 2.504

The green curve F(t) tracks exactly the area under f — F′(t) = f(t)

The Fundamental Theorem of Calculus is the result that makes calculus a unified subject rather than two separate collections of techniques. Before it, computing slopes and computing areas were distinct problems. After it, they are the same problem looked at from two directions.

Two problems, one theorem

In the 17th century, mathematicians were working on two apparently unrelated questions:

The tangent problem: given a curve, find the slope of the tangent line at a point.

The area problem: given a curve, find the area trapped beneath it.

Differentiation solved the first. Integration solved the second. The Fundamental Theorem says these operations are inverses — differentiation undoes integration, and integration undoes differentiation.

Part 1: the area function has a derivative

Fix a continuous function $f$ . Define the area function:

$F(x) = \int_a^x f(t)\,dt$

This is the area under $f$ from $a$ up to $x$ . As $x$ increases, $F(x)$ grows. How fast?

The answer is the FTC Part 1: $F'(x) = f(x)$ .

Why this makes sense: moving $x$ a tiny amount $\Delta x$ to the right adds a thin strip of area under $f$ . The strip has width $\Delta x$ and height approximately $f(x)$ , so $\Delta F \approx f(x)\,\Delta x$ . Dividing by $\Delta x$ and taking the limit: $F'(x) = f(x)$ .

The rate at which area accumulates under $f$ is exactly the height of $f$ at the current point. This is not obvious — it is the key insight.

Part 2: antiderivatives evaluate definite integrals

The practical consequence: to compute $\int_a^b f(x)\,dx$ , find any antiderivative $F$ (any function with $F' = f$ ) and evaluate:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Computing an area without summing rectangles

Before the FTC, computing $\int_0^1 x^2\,dx$ meant setting up a limit of Riemann sums:

$\lim_{n\to\infty} \sum_{k=1}^n \left(\frac{k}{n}\right)^2 \cdot \frac{1}{n} = \lim_{n\to\infty} \frac{n(n+1)(2n+1)}{6n^3} = \frac{1}{3}$

After the FTC: $\left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3} - 0 = \frac{1}{3}$ . Same answer in three seconds.

What the proof looks like

The proof of Part 2 from Part 1 is elegant. Define $G(x) = \int_a^x f(t)\,dt$ . By Part 1, $G'(x) = f(x)$ .

If $F$ is any other antiderivative of $f$ , then $(F - G)' = F' - G' = f - f = 0$ . A function with zero derivative everywhere is a constant. So $F(x) = G(x) + C$ for some constant $C$ .

Therefore: $F(b) - F(a) = [G(b) + C] - [G(a) + C] = G(b) - G(a) = \int_a^b f(t)\,dt$ .

The constant cancels. This is why any antiderivative works — they all differ by a constant, which disappears in the subtraction.

What it means philosophically

The FTC reveals that accumulation and rate-of-change are two sides of the same coin.

If you know how fast something is growing at every moment, you can reconstruct how much it has grown — just integrate the rate. If you know how much something has accumulated up to each moment, you can find the current rate — just differentiate.

This symmetry appears throughout applied mathematics:

Velocity is the derivative of position; position is the integral of velocity.
Current is the derivative of charge; charge is the integral of current.
Marginal cost is the derivative of total cost; total cost is the integral of marginal cost.

Every case where you move between "rate" and "total" is an application of the FTC.

The theorem that almost wasn't

Newton and Leibniz both recognised this connection in the 1660s–1670s, but it was not obvious to their predecessors. Archimedes could compute areas (imprecisely, what we'd now call definite integrals). Fermat could compute tangent lines (imprecisely, what we'd now call derivatives). Neither saw the connection.

The FTC did not just solve a problem — it reorganised how mathematics understood infinitesimal quantities. It is not an exaggeration to say that the modern scientific worldview, which uses differential equations to model everything from fluid dynamics to quantum mechanics, rests directly on this theorem.