The Fundamental Theorem of Calculus

Two problems opened this course (instantaneous rates and accumulated totals) and they have been developed separately. The Fundamental Theorem of Calculus is the bridge that joins them, and it is one of the great results in mathematics.

The idea

Imagine accumulating area under a curve as you move to the right, and let $A (x)$ be the area gathered up to the point $x$ . How fast does that accumulated area grow as $x$ advances a little? It grows by a thin sliver of height $f (x)$ , so the rate at which area accumulates is the height of the curve itself:

A' (x) = f (x) .

Accumulating (integrating) and then taking a rate (differentiating) cancel out. That is the first form of the theorem: the derivative of an integral gives back the original function.

Evaluating definite integrals

The second form turns this into a practical tool. If $F$ is any antiderivative of $f$ (any function with $F' = f$ ) then the area from $a$ to $b$ is simply the change in $F$ across the interval:

\int_{a}^{b} f (x) d x = F (b) - F (a) .

No rectangles, no limits of sums: just find an antiderivative and subtract its values at the endpoints.

A worked example

To find the area under $f (x) = x^{2}$ from $0$ to $3$ : an antiderivative is $F (x) = \frac{1}{3} x^{3}$ , so

\int_{0}^{3} x^{2} d x = \frac{1}{3} (3)^{3} - \frac{1}{3} (0)^{3} = 9.

The problem of summing infinitely many slivers collapses into a single subtraction.

Why it matters

The theorem is why the two branches of calculus are one subject. Every antiderivative you learn to compute becomes a way to measure an area, a volume, a total distance, an accumulated change. Differentiation breaks a whole into instantaneous rates; integration reassembles the rates into a whole; and the Fundamental Theorem guarantees the round trip brings you home.