The integral

Integration runs the derivative in reverse and, at the same time, measures area. That these are the same operation is the surprise at the heart of calculus.

Antiderivatives

An antiderivative of $f$ is any function whose derivative is $f$ . Since the derivative of $x^{2}$ is $2 x$ , an antiderivative of $2 x$ is $x^{2}$ . But so is $x^{2} + 5$ (the constant vanishes on differentiating) so antiderivatives come in families differing by a constant. We write the whole family as the indefinite integral

\int 2 x d x = x^{2} + C .

Area under a curve

The other face of integration is area. To find the area between a curve and the horizontal axis from $a$ to $b$ , slice the region into thin vertical rectangles, each of width $Δ x$ and height $f (x)$ at that slice. Each rectangle has area $f (x) Δ x$ , and their total approximates the region.

The definite integral

This sum of rectangle areas is a Riemann sum. As the rectangles grow thinner and more numerous, the approximation tightens onto the true area, and the limit is the definite integral

\int_{a}^{b} f (x) d x = lim_{Δ x \to 0} \sum f (x) Δ x .

The elongated $S$ is a stretched sum, Leibniz's reminder that an integral is, at bottom, an addition of infinitely many infinitesimal pieces.

Two integrals, one symbol

We now have two things wearing the same sign: the indefinite integral (a family of antiderivatives) and the definite integral (a number, an area). The next lesson explains why one symbol is right: they are bound together by the Fundamental Theorem.