Variational methods are closely connected to integrals. So, before starting our variational adventures, let us recall some elements about integrals and their numerical evaluation.
The history of infinite sums is such as integrals may be brought to Zeno of Elea’s paradox about the grain of millet. As described by Simplicius [FAI 98]:
Tell me, Protagoras, said he, does one grain of millet make a noise when it falls, or does the ten-thousandth part of a grain? On receiving the answer that it does not, he went on: Does a measure of millet grains make a noise when it falls, or not? He answered, it does make a noise. Well, said Zeno, does not the statement about the measure of millet apply to the one grain and the ten-thousandth part of a grain? He assented, and Zeno continued, Are not the statements as to the noise the same in regard to each? For as are the things that make a noise, so are the noises. Since this is the case, if the measure of millet makes a noise, the one grain and the ten-thousandth part of a grain make a noise.
Here, Zeno considers the problem of the sum of small negligible components, what is the principle of the integration of infinitesimal contributions. In these ancient times, the concept of the limit was not known, so the notion of infinite sums and the evaluation of areas remained unsolved. In fact, many philosophers considered the question of limits, such as Antiphon the sophist, who argued that continuously doubling the number of sides of a polygon ...