What are integrals, really?

Integral calculus is concerned with the computation of so-called integrals, such as:

which is in a way, an anti-derivative (e.g. the inverse operation of differentiating a function). In a broader sense, however, integrals represent measurable quantities, such as the area under a curve. For instance:

is the area beneath the parabola x² and above the x-axis between the two points x = a and x = b. Although we can integrate a lot of functions exactly, the fundamental principle behind a definite integral like this one is that we must start from simple rectangular areas of finite width Δx and height f(x) as shown in Figure 1 below:

Figure 1: Finite approximation of the area under a curve, representing the definite integral between points a and b.

The idea is that as we use more rectangles of narrower width, we get a more accurate approximation of the actual area, as shown in Figure 2:

Figure 2: Approximating areas using rectangles of different widths

The area (integral) can therefore be written as a sum of all these rectangles in the limit as the number of rectangles goes to infinity, namely:

which is called a Riemann sum. For the special (and very common) case where the rectangle widths are all the same, we write: