The Fundamental Theorem

Chapter 6.1 defined the integral as a limit of sums, which is honest and unusable — nobody evaluates $\int_0^2 x^2\,dx$ by summing rectangles. The Fundamental Theorem of Calculus is the shortcut, and calling it fundamental is not decoration. It says that the two central operations of the subject undo each other.

Part 2: evaluating integrals

Most-used first. If $F$ is any antiderivative of $f$ — that is, $F' = f$ — then

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

So $\int_0^2 x^2\,dx = \tfrac{2^3}{3} - \tfrac{0^3}{3} = \tfrac83$ — the number the rectangles were converging on, obtained in one line.

Any antiderivative works. $\tfrac{x^3}{3}$ and $\tfrac{x^3}{3} + 17$ are both antiderivatives of $x^2$, and the constant cancels in the subtraction: $(F(b) + C) - (F(a) + C) = F(b) - F(a)$. That is why the $+C$ that matters so much for indefinite integrals is irrelevant for definite ones.

Part 1: the derivative of an accumulation

Define a function by an integral with a variable upper limit:

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

Then $g'(x) = f(x)$.

Read that slowly, because it is the surprising half. Accumulating $f$ and then differentiating gives $f$ back. Integration and differentiation are inverses, and this is the precise statement of it.

Why it is true, in one sentence: $g(x + h) - g(x)$ is the area of a thin strip of width $h$ and height about $f(x)$, so the difference quotient $\tfrac{g(x+h) - g(x)}{h}$ is about $f(x)$, and exactly $f(x)$ in the limit.

That is the whole proof, and it explains the theorem rather than asserting it: the rate at which accumulated area grows is the height of the curve at the edge where it is growing.

With the chain rule

The upper limit is often a function of $x$, not $x$ itself. Then Part 1 composes with the chain rule:

$$\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f\big(u(x)\big)\cdot u'(x)$$

For $\displaystyle g(x) = \int_0^{x^2}\sin t\,dt$:

$$g'(x) = \sin(x^2)\cdot 2x$$

That verified line is the check: the integral evaluates to $-\cos(x^2) + 1$ by Part 2, and differentiating it gives $2x\sin(x^2)$ — the same answer the chain rule route produced, from an entirely different direction.

When the lower limit varies instead, the sign flips, because swapping the limits of an integral negates it:

$$\frac{d}{dx}\int_{u(x)}^{b} f(t)\,dt = -f\big(u(x)\big)\cdot u'(x)$$

The properties you get for free

All of these follow from the definition and are worth recognising rather than deriving under time pressure:

$$\int_a^a f = 0 \qquad \int_b^a f = -\int_a^b f \qquad \int_a^c f = \int_a^b f + \int_b^c f$$

The middle one is a convention, and it is the right convention: it makes the third hold even when $b$ lies outside $[a, c]$, which keeps the algebra uniform instead of full of special cases.

Linearity comes across from differentiation unchanged — integrals split over sums and let constants out front — because the sums defining them do.