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
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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