The Riemann Sum and Its Limit

The sum of the areas of the rectangles has a name: the Riemann sum. Writing $x_k$ for the subdivision points, it can be written compactly with the summation symbol:

$$S_n = \sum_{k=1}^{n} f(x_k)\,\Delta x, \qquad \text{with } \Delta x = \frac{b - a}{n}$$

$S_n$ is only an approximation of the area. But when the number of rectangles tends to infinity, their width $\Delta x$ tends to $0$, the overshoots and gaps vanish, and the sum converges to an exact value. This is the definition of the integral:

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{n \to +\infty} \sum_{k=1}^{n} f(x_k)\,\Delta x$$

This formula finally sheds light on the notation. The sign $\int$ is an elongated "S", for sum. The $\mathrm{d}x$ is what $\Delta x$ becomes in the limit: an infinitely small width. The integral is therefore, literally, a continuous sum of infinitely thin strips, of height $f(x)$ and width $\mathrm{d}x$.

Discrete sum	Integral, in the limit
$\displaystyle\sum_{k=1}^{n} f(x_k)\,\Delta x$	$\displaystyle\int_a^b f(x)\,\mathrm{d}x$
$n$ rectangles	$n \to +\infty$, so $\Delta x \to 0$

Remember the key point: the integral is not a magic formula, it is the outcome of stacking thinner and thinner rectangles. All of integral calculus rests on this idea.