The definite integral as a sum of differentials

In the chapter on derivatives we saw that the differential $df = f'(x)\,dx$ measures the infinitesimal change of a function. The integral is the inverse operation: it is an infinite sum of differentials.

Definition — The definite integral

The definite integral of $f$ from $a$ to $b$ is the sum of infinitely many infinitesimal rectangles of base $dx$ and height $f(x)$: $\int_a^b f(x)\,dx = \lim_{N\to\infty}\sum_{i=1}^N f(x_i)\,\Delta x, \qquad \Delta x = \frac{b-a}{N}.$ The symbol $\int$ is an elongated “S” (from summa); $dx$ is the infinitesimal “little piece” of the variable; $a$ and $b$ are the limits of integration.

Each little rectangle has base $dx$ (infinitesimal) and height $f(x)$: its area is $dA = f(x)\,dx$, a differential of area. The integral is the sum of all these differentials, “from $x=a$ to $x=b$”. Taking the limit $\Delta x \to 0$ and making the number $N$ of rectangles infinite, the sum becomes exact and coincides with the area of the region between the graph of $f$, the $x$-axis and the lines $x=a$, $x=b$.

The area beneath (in green) is approximated by many little rectangles of base $dx$ and height $f(x)$: the integral is the limit of their sum as $dx\to 0$.

Links

Topics: Integral
Concepts: Area beneath a curve · Differential · Definite integral · Riemann sum
Skills: Integrating