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This article is about the branch of mathematics. Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Archimedes used the method of exhaustion to calculate the area under a parabola.

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. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The formal study of calculus brought together Cavalieri’s infinitesimals with the calculus of finite differences developed in Europe at around the same time. Isaac Newton developed the use of calculus in his laws of motion and gravitation. Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.

These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, foundations refers to the rigorous development of the subject from axioms and definitions.