Who invented the first sentence of integration

integral

The integral is an important concept in mathematics. Besides differentiation, it is one of two main operations in calculus. Integral and differential calculus are inverse operations. That is, if you integrate a function f and differentiate it, you get the output function f again.Usually, integrated functions are written with capital letters (F).

Integrals are differentiated into certain integrals and indefinite integrals. A certain integral is defined as the area enclosed by the graph of the function f on the interval [a, b], with the vertical lines x = a and x = b serving as boundaries. The area above the x-axis has a positive sign, while the area below the x-axis is subtracted from the total area.

However, integration can also be defined as the inverse operation for differential calculus. In this case the integral would be the Indefinite integral a function f and thus a indefinite integral. The antiderivative is not defined on an interval.

The principles of integration calculation were formulated independently of one another by Sir Isaac Newton and Gottfried Leibniz in the late 17th century and were originally defined as an infinite sum Rectangles of infinitely small width. A more precise mathematical definition of the term integral was made by Bernhard Riemann in the 19th century.
Integrals play a central role, especially in differential geometry. The first generalizations of the term integral were driven by physics, in which integration plays an important role in many physical laws, especially in electrodynamics.

Historical development of integral calculus

The first documented mathematical method for calculating areas, i.e. integration, was that Exhaustion method, developed by the Greek astronomer Eudoxus of Knidos (approx. 370 BC). The ancient Greek philosopher Antiphon was convinced that the circle could be quartered, since any other polygon can be converted into a square. He assumed that a polygon with a certain number of sides would be identical to a circle. On the basis of this consideration, Eudoxus developed the exhaustion method. The unknown area of ​​any figure or any polygon can be determined mathematically by filling its area with polygons whose area calculation is known. If the number of these polygons is converted to infinity, their area becomes infinitely small while their number becomes infinitely large. This makes the difference between the area of ​​the polygons and the area of ​​the figure infinitely small. Archimedes developed this method further in the third century BC to approximate the areas of parabolas and the circle.

The next milestone for integral calculus was made by the Italian mathematician Bonaventura Cavalieri in the 16th century. He discovered with the one named after him Principle of Cavalieri, that polygons (in two-dimensional space) and figures (in three-dimensional space) are the same under certain circumstances. He also calculated the integrals of xn up to n = 9.

The first indications that there is a connection between integral and differential calculus were made at the beginning of the 17th century by Torricelli and Barrow. Barrow provides the first proof of the fundamental theorem of calculus. The English mathematician John Wallis extended Cavalieri's formula to arbitrary powers (including negative numbers and fractions).

Leibniz and Newton

Gottfried Leibniz and Sir Isaac Newton discovered this independently Fundamental theorem of analysis. The theorem establishes the connection between integral calculus and differential calculus. This connection, along with the fact that derivatives are relatively easy to compute, can be used to compute integrals in turn. The work of Leibniz and Newton represents the basis of modern analysis, whereby the notation for integrals was introduced by Leibniz and is still used today.

Terminology and notation

  • integral
    The notation for the integral as we use it today was originally invented by Gottfried Wilhelm Leibniz. It should represent a stylized "S" (for "sum") and express that we add the sum of the area of ​​an infinite number of rectangles (Riemann integral), all of which have an infinitely small width.
  • Upper and lower limit
    The upper and lower limits are only important for certain integrals. The upper and lower bounds do not have to be numbers. Variables, terms or ± ∞ are also possible. If the integration limits are specified, one speaks of adefinite integral. On the other hand, an integral without an upper or lower limit is calledindefinite integral.
    If infinity is specified as the integration limit, it is possible that the result of the integration tends to reach a certain value. Here it is usually necessary to consider the limit value!
  • Integrand
    The integrand is the function to be integrated. Even if the integrand is mostly a function of the integration variable, this does not necessarily have to be the case.
  • differential
    The differential has historical significance. Let us take the Riemann integral as an example. Here rectangles are used to calculate the area between the curve and the x-axis. The smaller the width of the rectangles, the more accurate the result of the Riemann integral. The d indicates exactly this: it tells us that we have to make the width of the rectangle almost infinitely small.
  • Integration variable
    The integration variable indicates which variable is important for the process of integration. It is important to note the integration variable as it is not always x. In physics and other natural sciences in particular, other variables such as t for time or r for radius are often used.

Definite integral

If the integration limits are given for an integral, it is called definite integral. After the antiderivative has been found, the upper and lower limits must be inserted and a value calculated. This value corresponds to the area between the function and the x-axis in the interval [a, b]. If the function runs below the x-axis, the result is negative.

A certain integral is calculated like this:

After the antiderivative has been determined, the upper limit and lower limit are inserted and subtracted from each other. This is also known as the second law of analysis.

Negative area

The definite integral calculates the area of ​​a function between the lower and upper limit of the integral. It is better to speak of the net area, since the area becomes negative if the function is subtracted from the total area below the x-axis and when it is integrated. Let's look at a simple example:

The antiderivative of the function is . So the definite integral from 0 to 1 of f would be the same . As you can see from the graph (right), the graph of the function f (x) = x for values ​​less than zero is below the x-axis. Since the function is point-symmetrical to the origin, the amount of the area, starting from the origin, is identical (only the sign is different). Therefore the integral from -1 to 1 is zero:

Therefore, if one wants to calculate the absolute area, one must first determine the zeros of f (x), and then form an integral from the lower limit to the zero and from the zero to the upper limit. Since the area can also be negative, we add the amount of the sums. So the absolute area would be:

Indefinite integral (antiderivative)

The indefinite integral (also Indefinite integral called), can be seen as the reverse of differentiation. Since the derivative does not completely determine the function, we add "+ C" to the antiderivative (you can add any constant to an output function f and its derivative will remain the same). this is the Constant of integration.

In contrast to the definite integral, the antiderivative is not determined on an interval, but in general, the function that determines the area between the x-axis and the graph. The antiderivative is usually the starting point for calculating the area.

In addition, it should be noted that we are mostly from surface speak, but this is not necessarily correct depending on the context and the question. From a physical point of view (and thus an application-oriented point of view) one rarely looks for a surface when integrating.