Integral calculus
Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Moscow papyrus (c. 1820 BC), in which an Egyptian mathematician successfully calculated the volume of a pyramidal frustum.
Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.
•Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes' Quadrature of the Parabola, The Method, Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true.
In the third century Liu Hui wrote his Nine Chapters and also Haidao suanjing (Sea Island Mathematical Manual), which dealt with using the Pythagorean theorem (already stated in the Nine Chapters), known in China as the Gougu theorem, to measure the size of things. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, showing a grasp of elementary concepts associated with the differential and integral calculus. In the 11th century, the Chinese polymath, Shen Kuo, developed 'packing' equations that dealt with integration.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first written by Aryabhata circa 500 AD, in order to find the volume of a cube, which was an important step in the development of integral calculus.
The next major step in integral calculus came in the 11th century, when Ibn al-Haytham (known as Alhacen in Europe), an Iraqi mathematician working in Egypt, devised what is now known as "Alhazen's problem", which leads to an equation of the fourth degree, in his Book of Optics. While solving this problem, he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.
Parallel development
In the 17th century, Pierre de Fermat, among other things, is credited with an ingenious trick for evaluating the integral of any power function directly, thus providing a valuable clue to Newton and Leibniz in their development of the fundamental theorem of calculus. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.
At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly Kowa Seki. He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion. While these methods of finding areas were made largely obsolete by the development of the fundamental theorems by Newton and Leibniz, they still show that a sophisticated knowledge of mathematics existed in 17th century Japan.
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دانلود پاورپوینت History of calculus