Many quantities are obtained by adding contributions spread continuously over an interval: area, distance, mass, charge and many others. A definite integral describes this kind of total accumulation.

The notation

$$\int_a^b f(x),dx$$

represents the signed accumulation of $f$ from $x=a$ to $x=b$.

Area as a first example

If $f(x)\ge0$ on $[a,b]$, the definite integral is the area between the graph of $f$ and the $x$-axis. Regions below the axis contribute negatively, so an integral measures signed area, not simply geometric area.

For a constant function $f(x)=c$,

$$\int_a^b c,dx=c(b-a),$$

which is exactly the signed area of a rectangle.

Riemann sums

For a varying function, divide $[a,b]$ into many short subintervals. A rectangle of width $\Delta x$ and representative height $f(x_i^*)$ contributes approximately

$$f(x_i^*)\Delta x.$$

Adding the rectangles gives a Riemann sum:

$$\sum_{i=1}^n f(x_i^*)\Delta x.$$

As the partition becomes finer, these sums may approach a limit. The definite integral is defined by that limiting accumulation.

The notation

In

$$\int_a^b f(x),dx,$$

$a$ and $b$ are the bounds of integration, $f(x)$ is the integrand, and $x$ is the integration variable. The symbol $dx$ also reflects the idea of accumulating contributions over very small changes in $x$.

Discovering integration from accumulation