Chapter 8 The Mechanical World: Descartes and Newton 5. Apply the geometric techniques of Wallis to obtain the volume of the solid generated by revolving the region beneath the curve y x and above the...

Question 4

Extracted text: Chapter 8 The Mechanical World: Descartes and Newton 5. Apply the geometric techniques of Wallis to obtain the volume of the solid generated by revolving the region beneath the curve y x and above the x-axis, with 0 xSa. [Hint: Consider the solid to be made up of 410 (b) small rectangles" to find the area under the curve y = xover the interval [0, a]; in integral notation this amounts to calculating 'dx. Use Wallis's method of partitioning by "infinitely n circular disks of width and radii (4)2. Add becomes infinitely large. In doing so find 4 s**** their volumes and take the limit as na up dx, obtain xdx. 4. Given Wallis's value for y /y = x2 04+ 14+24+34++n L lim n4 +nt +nt + +nt (а, а) NT dx from the formula n(n + 1)(2n + 1)(3n2+3-1 14+2+34+n X 30 va The invention of the calculus was one of the great tellectual achievements of the 1600s. By one of those curious coincidences of mathematical history not one Gottfried Leibniz: The Calculus Controversy 8.4 but two men devised the idea-and almost simul neously. The methods of the calculus of Newton i England and Leibniz on the Continent were so sul that the question whether Leibniz borrowed the crucial concepts from Newton or discovered them independently gave rise to a long and bitter controversy. The tactics of the principal protagonists were so unworthy of these two titans, and the violence of the accusations and The Early Work of Leibniz counteraccusations so injurious, that neither escaped with his reputation untarnished. When inferences of plagiarism became public charges, a committee of the Royal Society, called to adjudicate this most notorious of scientific disputes, found-not surprisingly-in faver of the society's own president against one of its oldest foreign members. Gottfried Wilhelm Leibniz (1646-1716) was born in the university town of Leipzg some two years before the Peace of Westphalia put an end to the Thirty Years' War. His father, a jurist and professor of moral philosophy at the university, died when the boy was 6 years old. As a result, the young Leibniz was left almost without direction in his studies The boy's world was the world of books. A precocious child, he taught himself Latin from an illustrated copy of Livy's history of Rome when he was about 8, and had begun study of Greek by the time he was 12. This led to his being given unhampered access 0 his father's library, which had previously been kept under lock and key. Here, according his own testimony, he became acquainted with a wide range of classical writers. Leihni wrote in later life: "I began to think when I was very young; and before I was fifteen Iused to go for long walks by myself in the woods, comparing and contrasting the principles af the Aristotle with those of Democritus." In the fall of 1661, the same date that Newton entered Cambridge, Leibniz became student at the university of his native city, Leipzig. Only 15 at the time, he was regardeds something of a prodigy and soon outstripped all his contemporaries. The education received at Leipzig followed traditional (orthodox Lutheran doot Leibais relig