Secondary special education of the republic of uzbekistan

Egypt and Babylonia. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter.

Greece. From the age of Greek mathematics, Eudoxus (c. 408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287–212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also 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 seemingly create.

Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola.
Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his The Quadrature of the Parabola, The Method, and On the Sphere and Cylinder. It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into a general framework of integral calculus. Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983).


Archimedes used the method of exhaustion to approximate the value of pi.

Apollonius of Perga made significant advances in the study of conic sections.
China. The method of exhaustion was independently invented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Liu Zhuo was the first to use second-order interpolation for computing the positions of the sun and the moon.

Counting rod numerals.