A History of Calculus, Pt. 3 – Integrals

Development

In calculus courses, integral studies follow derivative studies, a natural sequence given that integration is, in essence, the reverse of differentiation. However, in mathematics discovery did not follow this sequence.

In the days of antiquity, scholars fiddled more with the ideas from which integral calculus would branch than those from which differential calculus would. We start out journey through integral history in ancient Greece, where the eminent scientist Archimedes adapted a technique from past scholars to develop the method of exhaustion, which measured the circumference of a circle.

The method of exhaustion (Image from Wikimedia Commons)

Using the method of exhaustion, Archimedes found the circumference of the circle we know today, P = 2πr. He accomplished this by circumscribing a regular polygon outside a circle and inscribing a regular polygon in it, reasoning that the true circumference of the circle was less than the perimeter of the circumscribed polygon and greater than the perimeter of the inscribed polygon. ^[1] As the number of sides of the polygon increases, the calculated perimeter draws closer and closer to the actual circumference of the circle. In these golden days of mathematical discovery, we could say that the perimeter of a circle is equal to the perimeter of a regular polygon of n sides as n approaches infinity. This same process can be applied to finding the area.

Non-European mathematicians also contributed to what would become integral calculus, notably among them ibn al-Haytham of Arabia (965-1040), who found a way to calculate the volume of a solid by thinking of it as the area under a curve that we rotate around an axis, and that resultant shape as the infinite sum of infinitely thin disks. ^[2] (In modern calculus terms, we call this the disk method of integration.)

An example of the disk method (Graph from Montessori Muddle)

As Europe trudged through the Middle Ages, scholars in the East steamed ahead with innovative, mathematical constructs, among them higher-degree polynomials and, consequently, sums of powers, the method used to calculate area until the invention of the integral.

Both Newton and Leibniz acknowledged integrals as the inverse of derivatives _[3]. They both alluded to a Fundamental Theorem of Calculus, which demonstrated this connection. However, not until French mathematician Augustin-Louis Cachy (1789-1857) was the theorem formally expressed.

The integral represents in a nice, neat package that which mathematicians long wanted to calculate – the area under under a curve and the volume of a rotated graph. In the late 17th century, armed with Archimedes’ method of exhaustion, the Eastern mathematicians’ polynomials, and Newton’s and Leibniz’s ground-breaking work in calculus, Bernhard Riemann entered the scene.

Progression of Riemann sums (Image from Wikimedia Commons)

To notate integration, Leibniz invented the integral notation known today, the elongated S, alluding to the intuitive understanding of the integral as the infinite summation of infinitesimally small parts. Riemann demonstrated this idea by interpreting the area under a curve as the sum of an increasingly greater number of increasingly thinner rectangles contained under the curve. This is Riemann summation. If one has n rectangles under the curve of Δx width, one will draw closer and closer to the exact area as n approaches infinity and Δx approaches 0.

This limit, we find, is the integral.

Definition

The integral has two main definitions:

1. The inverse of a function’s derivative, and
2. The area under a curve

As far as the first definition goes, just consider the function f(x) = 2x². Its derivative is 4x. The integral of 4x is the function for which 4x is the derivative, i.e. the original f(x).

For the second definition, think about this: If we have the derivative f'(x), the integral is the sum of the areas at each point under the curve of f(x), which would give us the area under the whole curve.

The limit definition of an integral (Image from Statistics How To)

(Note: The left side of the equation uses Leibniz’s integral notation, with a and b being the lower and upper bounds of the function in question. These bounds give us a definite integral – a contained area. An integral without bounds is an indefinite integral. The jerky E on the right side is the Greek letter sigma, which represents a mathematical sum, the basic idea behind Riemann sums.)

Other posts in “History of Calculus”

• Part 1 – Limits
• Part 2 – Derivatives
• Part 4 – Applications

[1] Archimedes’ method of exhaustion

[2]Calculus before Newton and Leibniz

[3] A brief history of the integral