Definite integral

Definite integral DEFINITE INTEGRAL Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function Æ’ of a real variable x and an interval [a, b] of the real line, the definite integral, is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of Æ’, the x-axis, and the vertical lines x = a and x = b. APPLICATIONS OF DEFINITE INTEGRAL Definite integrals arent just for area any more Any definite integral may be interpreted as a signed area. Area, volume, arc length, work, mass, fluid pressure, and accumulated financial value are quantities that may be calculated with definite integrals. The most important components of these problems are constructing the correct integral and Interpreting the results.n TWO VIEWS OF DEFINITE INTECRAL When using the definite integral to solve various problems, it is useful to consider two different interpretations: A limit of approximating sums: The definite integral is formally defined as a limit of approximating sums using right sums. Accumulated change in an antiderivative: The Fundamental Theorem of Calculus states where F is any antiderivative of f on [a ; b]. The difference F(b) F(a) represents the accumulated change (or net change) in F over the interval [a; b]. To find the accumulated change in F over [a; b], integrate f, the rate function associated with F, over the interval [a ; b]. WHICH VIEW IS BETTER : SUM OR ANTIDERIVATIVE ? Often we need to decide which view (or interpretation) of the definite integral is the correct one for a given application. It could be that an approximating sum is acceptable or that a precise symbolic antiderivative is more appropriate. If an integral is presented in symbolic form, then antidifferentiation seems reasonable. For data given graphically or in a table, approximating sums are the logical choice. EXPLANATION: TRAPEZOIDAL RULE The trapezoidal rule (also known as the trapezoid rule, or the trapezium rule in British English) is a way to approximately calculate the definite integral The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that To calculate this integral more accurately, one first splits the interval of integration [a,b] into n smaller subintervals, and then applies the trapezoidal rule on each of them. One obtains the composite trapezoidal rule: Illustration of the composite trapezoidal rule (with a non-uniform grid) This can alternatively be written as: Where (one can also use a non-uniform grid). The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas. Simpsons rule is another, often more accurate, member of the same family. Simpsons rule and other like methods can be expected to improve on the trapezoidal rule for functions which are twice continuously differentiable; however for rougher functions the trapezoidal rule is likely to prove preferable. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler–Maclaurin summation formula. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurat ely SIMPSON RULE In numerical analysis, Simpsons rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation: Simpsons rule can be derived by approximating the integrand f(x) (in blue) by the quadratic interpolant P(x) (in red). METHODS BASED ON UNDETERMINED COEFFICIENTS NEWTON-COTES METHODS: TRAPEZOIDAL METHOD We have n=1 , x0 =a , x1=b and h=x1-x0. Rn= (1) Using eq 1 ,the rule can be made exact for polynomial of degree upto one.For f(x)=1 and x, we get the system of equations . f(x)= 1: x1- x0 = + or = + f(x) = x: Â ½ ( ) = + ( ) ( ) = + h( 2 + h ) = + ( ) h( 2 + h ) = ( + ) + h = h + h h= , or = From the first equation , we get h = h /2 . The method becomes = [ f( ) + f (] The error constant is given by C = [ ] [ ] = [ 2 ( + 3 h + 3 + ) -2 -3 h -3h( + 2h + ) ] = SIMPSON` S METHOD We have n = 2 , = a , = + h , = + 2h = b , h=(b a )/2 .We write = f( ) + f() + f( ) The rule can be made exact for polynomials of degree upto two . For f(x) = 1, x , , we get the following system of equations. f(x) = 1: = + + , or 2h = + + (2) f(x) = x: ( ) = + + -(3) f(x) = : ( ) = + + (4) From (3) , we get ( ) ( ) = + + h) + + 2h) (2h) (2+ 2h) = ( + + ) + ( + 2 ) h = 2h + ( + 2 ) h 2h = + 2 (5) From (4) , we get [( + 6 h + 12 + 8 ) ] = + ( + 2 h + ) + ( + 4 + 2 h + ) h + ) Or h = + 4 (6) Solving (5) , (6) and (2) , we obtain = , = , The Method is given by .., = [ f() + 4 f() + f () The error constant is given by C = = COMPARISON BETWEEN TRAPEZOIDAL RULE AND SIMPSONS RULE Two widely used rules for approximating areas are the trapezoidal rule and Simpsons rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical scheme will follow. This is the idea behind the trapezoidal and Simpsons rules. Here the trapezoidal rule approximates the function by a suitably chosen (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpsons rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points. The Maple student package has commands trapezoid and simpson that implement these methods. The command syntax is very similar to the rectangular approximations. See the examples below. Note that an even number of subintervals is required for the simpson command and that the default number of subintervals is n=4 for both trapezoid and simpson. > with(student): > trapezoid(x^2,x=0..4); > evalf(trapezoid(x^2,x=0..4)); 22 > evalf(trapezoid(x^2,x=0..4,10)); 21.44000000 > simpson(x^2,x=0..4); > evalf(simpson(x^2,x=0..4)); 21.33333333 > evalf(simpson(x^2,x=0..4,10)); 21.33333333 EXAMPLES OF TRAPEZOIDAL AND SIMPSON`S RULE Ques:Evaluate using trapezoidal and Simpson`s Rule with h=0.05 Sol: x0= 1 , x1= 1.05 , x2= 1.1 , x3= 1.15 , x4= 1.20 , x5=1.25 , x6= 1.3 I(trapezoidal) = = .05/2[ f(1) + 2( f (1.05) + f(1.1) +f(1.15)+ f(1.120) +f (1.25)) +f(1.3)] = 0.326808 = = = I(simpson) = [f(1) + 4 (f (1.05)+ f(1.15) + f(1.25) + 2(f(1.1) + f(1.20) +f(1.3) ] = 0.321485 Ques 2 :Find the approximate value of I= Using (i) trapezoidal rule and ,(ii) Simpson`s rule.Obtain a bound for the error. The exact value of I=ln2=0.693147 correct to six decimal places. Sol: Using the Trapezoidal rule , I= ( 1+ ) = 0.75 Error = 0.75 0.693147 = 0.056853 Using the Simpson`s Rule, I= (1+ + ) = = 0.694444 Error = 0.694444 0.693147 = 0.001297

Public and Private Policing Essay

Police, Securityguard, Security, Crimeprevention, Constable, Criminaljustice, SurveillancePublic and Private Policing The growing privatization of police services is a global phenomenon. It was first widely noted in the United States in a 1972 Rand Corporation study commissioned by the National Institute of Justice. Several years later, Stenning and Shearing observed that a â€Å"quiet revolution† toward private policing had occurred in Canada. South documented a similar trend in both western and eastern European countries. And an update of the original Rand assessment in 1985 concluded that private security outspent public law enforcement by 73 percent and employed two and a half times as many people. Public and private policing have many similarities, as well as differences and the distinction between public and private police are often blurred. Private policing, while emerging as a new industry, is not a new phenomenon and predates the existence of public police as witnessed today (Wilson 1994, p. 285). There are at least three reasons for the dramatic increase. First, in both post-industrial and developing nations, there has been an increase in what Stenning and Shearing call â€Å"mass private property†: shopping malls and gated communities. These spaces have traditionally fallen outside of the domain of public police, although this is now changing. Second, the fear of crime among those with property has grown faster than government’s willingness to spend more money on police protection. In many countries, this fear of crime among the propertied classes was intensified by the transition from authoritarian to democratic rule. Third, private police forces have often placed a higher priority on visible patrol than public police, hoping to deter crime through their presence. As early as 1971 Scott and McPherson worried that private policing might infringe upon civil liberties with impunity. Formal and familiar mechanisms exist around the world to hold public police accountable for their actions, but accountability mechanisms for private police are less well understood and often emanate from private rather than public institutions. In many cases, the state has little power or incentive to hold private police accountable. Stenning, however, believes that the inadequate accountability of private police has been overstated; marketplace competition, consumer pressures, demands of organized labor, and potential civil liability, he argues, compensate for lesser state regulation and oversight. Public policing has been known to have a monopoly on policing until the increased trend of private policing in the United States. Public police consist of the governmental department charged with the regulation and control of the affairs of a community, now chiefly the department established to maintain order, enforce the law, and prevent and detect crime. Private policing refers to that policing activity of crime prevention, detection and apprehension carried out by private organizations or agents for commercial purposes. Private policing may be defined to include those people who work for a security company or are employed by an individual or firm to carry out security work, crowd control or private investigations. In seeking to describe the policing activity of private police, however, most functional definitions stem from the perceived role of the public police (Nalla & Newman, 1990). Private police look and behave like public police and describing their function often involves a comparison of the activities and responsibilities of the two. Despite the differences, public and private police tend to mirror each other to a certain extent (Nalla & Newman, 1990). Private policing is provided by a private individual or organization, rather than by a public body or the state like public policing is. Private police are seen to be concerned with the protection of personal and corporate interest while public police represent the interests of the public and seek to enforce the regulations of the judicial system. The police are â€Å"persons with a special legal status employed by governments to preserve the peace† (Shearing, Farnell & Stenning, 1980,) Private policing, in comparison to public policing, has been described as passive policing as to active policing, or as proactive and preventative rather than reactive: where public police generally react to the crime, private police through surveillance and presentation are seen to prevent crime. (Wilson 1994) Private policing targets private crime and is in the business of protecting private and corporate interests. Private policing usually operates behind the traditional and legal boundaries where the public police cannot lawfully cross unless by invitation or probable cause. This leads to the private policing sector having a â€Å"broader enterprise than public policing, with a wider range of functions. † (South, 1988, p. 4) One difference between public and private police is private investigators are hired by individuals or businesses for a certain purpose and work mostly behind the scenes or undercover doing surveillance while Public policing is known to society as the police who protect our communities and arrest those individuals who commit crime. Public policing has the role to maintain law and order, preserve peace and prevent crime. Public police are employed by governments and paid for by tax dollars and grants. Another difference between the role of public and private policing is the private providers of security is their flexibility. They can, and will, perform most tasks they get paid to do. Their customers can demand a lot from them, since they are directly answerable to the paying clients and their needs. The private entrepreneurs are also forced to ‘do right’ by the market. If they fail, they will lose their money. Public police do not have the negotiation factor and are paid on salary, no matter how they perform or how efficient they are in performing their duties. Many have said that private policing is for the rich and public policing is for the poor. This could be effectively argued based on the fact that private policing is not designed to consider the general good for society, like public policing. Private policing is primarily protecting the interests of their paying clients and focuses more on â€Å"loss prevention†, rather than â€Å"crime prevention†. Private policing has been scrutinized and concern has expressed that private security can be overly intrusive, less than scrupulous in its adherence to self-imposed guidelines and, on occasion, the law, and threatening to civil liberties. Although public and private both play a major role in society, they do have different responsibilities. The responsibilities of a public officer include preserving the peace, preventing crimes and other offenses, assisting victims of crime, apprehending criminals, laying charges, prosecuting and participating in prosecutions, executing warrants, performing the lawful duties assigned by the chief of police, and completing the required training. The responsibilities of a private officer include training for private investigators and security guards is generally the responsibility of the employer. No license is required if the private investigator or security guard is hired . in house,. which means that he or she is an employee of, for example, an insurance company, court house, law firm, or store. Compared with police officers, private security in Canada is characterized by the following: lower wages, minimum or no recruitment standards, higher percentage of part-time work, higher turnover rate, lower levels of education, and minimum or no training (Marin,1997). In conclusion, Public and private policing are major components in the criminal justice field. It would be impossible for our communities to feel secure without the combination of both forces. The focus has been on public and private policing to effectively interact and cooperate with each other. Understanding the importance of one another’s responsibilities and roles could lead to a great partnership. Throughout recent years, some law enforcement agencies have come to realize how to benefit from private policing. Private security personnel differ from police officers in a number of ways. Private security personnel work for clients who pay them for services rendered, while police officers are responsible for serving and protecting the public. Minimum requirements and training are considerably less for private security than for police officers.

Asian Cultural Heritage

ASIAN CULTURAL HERITAGE The popular and widely known Asian â€Å"Migration Theory† of professor H. Otley Beyer about the prehistoric peopling of the Philippines is now considered untenable by modern scholars. They questione the manner by which the theory was formulated because of the absence of historical or archeological proofs to support it. The Asian texture of Filipino culture from the prehistoric Spanish times, therefore, could be traced to evolutionary factors, the continous migrations of people and the early contacts with other Asian civilizations. INDIAN INFLUENCESIndian influences were traceable in the languages, religious belief, literature, customs and traditions of early Filipinos. The Supreme God of the ancient Tagalog was Bathala, which came from the Sanskrit word B’hattara (great Lord) Among the pre-Islamic natives of Sulu, Indra Battara was the most prominent deity, Indra, being the sky god. The other gods the natives believed were of Indian or Vedic in origin such as Agni (fire god) and Surya (sun god). The epics of the early Filipinos such as the Biag ni Lam-Ang of the ilocanos, Mahabharata, the Indian epic.Some Filipino superstitious beliefs that originated from India were: 1. A comet is a bad sign; it brings war, famine and other calamities. 2. A pregnant woman must not eat twin bananas for she will give birth to twins. The putong (headgear) of the early Filipino male was Indian origin. So was the sarong (Indian sari), the lower part of the clothing of pre-Spanish women. Indian influences on Filipino culture are clearly manifested by the presence of Sanskrit words in Tagalog language. Dr. T. H. Pardo de Tavera mentions 340 Sanskrit words. Examples are: SanskritTagalog . Atawaasawa (spouse) 2. Amaama (father) 3. Raharaha (king) 4. Harihari (king) 5. Kottakuta (fort) 6. Gandaganda (beauty) 7. Bhattarabathala (god) 8. Mutyamutya (pearl) 9. Inaina (mother) MALAYAN INFLUENCES The Maragtas For lack of historical or archeological data , the Maragtas or the great tale about the Ten Bornean Datus who came to Panay may be part-history and part-fiction. According to this great story, at around 1250 AD, ten datus and their families left Borneo to escape the repressive rule of Sultan Makatunaw and to establish new homes across the seas.Led by Datu Puti, the Malays landed in Panay Island where they negotiated with Marikudo, the Negrito king for the barter of the lowlands. The agreed price was one golden salakot for the Ati king and one gold necklace for Maniwangtiwang, Marikudo’s wife. The barter of Panay was sealed by an agreement of friendship between the Atis and the Malays. The Negritos, after performing their merry songs and dances, retreated to the mountains. There are some interesting coincidences, however, between the legend of Maragtas and Panay customs and traditions.To this day, the ati-atihan, a colorful song and dance festival is celebrated in Aklan. This is to re-enact the warm welcome accorded to t he Malays by the Atis, which resulted to the purchase of Panay. W. H. Scott, the American missionary, in his doctoral dissertation (1968) comments: â€Å"There is no reason to doubt that this legend (Maragtas) preserves the memory of an actual event, but it is not possible to date the event itself, or to decide which of the details ar historical facts, and which are the embellishments of generations of oral transmission. CHINESE INFLUENCES The Chinese, who came to the Philippines, whether as traders or settlers, were primarily interested in trade, so their influences on Filipino life were mainly economic and social. The early Filipinos learned from the Chinese the art of metallurgy, the manufacture of gunpowder, mining methods and the use of porcelain, gongs, umbrellas, lead and kites. From the Chinese originated the dishes lumpia, mami, okoy, pansit, bihon, chop suey and siopao. Sauces like toyo, and tawsi also came from the Chinese.Some Chinese customs were eventually adopted by the Filipinos. The arrangement of marriage of children by parents, the use of go-between in negotiating marriage, the use of white clothes or dress during the period of mourning and the filial respect for elders accorded by the children were examples of practices borrowed from the Chinese. About a thousand words are found in the Filipino language. Among them are the following: ChineseFilipino 1 Inkongingkong 2 Hebihibi 3 A-chiate 4 Bi-koebiko 5 Dikiamdikiam Pin-topinto 7 Sosisusi 8 Mikimiki 9 Bakkiahbakya 10 Pansitpansit JAPANESE RELATIONS The Filipinos had been trading with the Japanese long before the coming of the Spaniards. Japanese bahan (merchants) and wakos (pirates) sailed the South China Sea seeking for Sung and Yuan wares buried in Philippines graves, iron and woolens for Filipino gold and wax. The Japanese immigrants who came to the Philippines during the pre-Spanish times settled at the mouth of the Cagayan River, the Lingayen Gulf area, and Manila.The first recorded enc ounter between the Japanese and the Spaniards was in 1572 when Juan de Salcedo, while sailing from Manila to Ilocos, fought off three Japanese junks off the coast of Pangasinan. The Japanese pirates sailed away, after fierce fighting. In 1582, an expedition led by Captain Juan Pablo Carreon assaulted a Japanese colony founded by the pirate Tayfusa at the mouth of Cagayan River. The Japanese were forced to leave the place. The early Japanese merchants also traded with Agoo, a town in the Lingayen Gulf area.They brought with them utensils, assorted weapons, salted meats and other Japanese were better treated and nodiscrimatory decrees were made against them. The Spaniards respected and feared them. ARABIC INFLUENCES Arabia’s most enduring legacy to the peoples of Sulu and Mindanao is Islam. The Arabs also introduced the sultanate form of government, Arabic art and literature; the Arabic alphabet; the Koranic Law; the mosque; the art of warfare- lantaka(cannon), vinta(warboat) a nd the kuta(fort).The Luwaran, a code of laws and compilation of the customs and traditions of the early Muslims, was written in Arabic. The Maranao’s Darangan (epic poetry), Maguindanao’s Indarapatra and Sulayman and Sulu’s Parang Sabil were deeply inspired by Islam. Arabic influence is also clearly seen in the decorative and ornamental art of the maranaos, who are the best –known wood carvers and painters of the region. Arabic words are also found in the Filipino language. Examples are: surat(letter), apu(old man), akma (appropriate), arak (wine), alamat (legend), maalem (knowing), pirate (scar) and salam (thanks).

The Country Of The United States - 1268 Words

A country inhabited by over six million people off of the coast of Central America. A ravishing land full of beaches and wildlife, but also a place ravaged by corruption and gang populations that feel that they are leaders of the urban areas. This country has been torn by the result of war games played among its own brothers. The beautiful nation of El Salvador that has been the place of birth for my parents, grandparents and ancestors. San Salvador is the nation’s capital and possibly most prominent area of my land. The smallest of all Central American countries yet the most densely populated. Our flag wave’s two streaks of blue and one of white topped by a symbol representing the coat of arms in the center. Salvadorian people also being†¦show more content†¦It was by many dire battles and the multitudinous amount of troops available to Spain alongside financial and advanced weapon aid that the Spaniards were eventually able to conquer this territory. San Salv ador is one of the first cities established by the Spaniards, they moved rapidly to colonize and change the language of those under their rule. From many different tribal tongues to the Spanish that is now part of home life for the majority of all Central American countries. The freedom of El Salvador and all of Central America from Spanish rule came on September 15, 1821. El Salvador also led Central America in the movement to liberate itself from Mexican-Spanish rule by evading taxation and resisting development efforts that would destroy the way of life and cultures of those native Salvadorians, all concluding in an uprising by the Salvadoran people well until the collapse of the Mexican empire. A big part of Salvadoran life is the beliefs placed upon by our elders. The countries most followed religion is that of the Roman Catholic Church, with a following of around fifty-six percent of all Salvadorans; however twenty-one percent of people identify as Protestants. This leaves three percent of the population converting to other religions. Religion is so heavily followed by the