Definite integrals were defined through accumulated area, while antiderivatives were defined by reversing differentiation. The fundamental theorem of calculus shows that these two ideas are two sides of the same process.

Accumulation creates an antiderivative

Let $f$ be continuous and define

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

As $x$ changes by a small amount, the accumulated area changes by approximately $f(x)$ times that small width. In the limit,

$$F'(x)=f(x).$$

So accumulating a continuous function produces an antiderivative of that function.

Evaluating a definite integral

If $G$ is any antiderivative of $f$, so that $G'=f$, then

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

For example,

$$\int_0^2 3x^2,dx =\left[x^3\right]_0^2 =8.$$

This replaces a limiting sum of many small contributions with two evaluations of an antiderivative.

Why the constant disappears

Any two antiderivatives differ by a constant. If $G(x)+C$ is used instead, then

$$(G(b)+C)-(G(a)+C)=G(b)-G(a),$$

so the definite integral is independent of that constant.

The theorem is the central bridge between differentiation and integration.

Discovering the derivative of accumulated area

Integration and the fundamental theorem