title: M1 carbine

text: During World War II, the British SAS used the M1 and M1A1 carbines after 1943. The weapon was taken into use simply because a decision had been taken by Allied authorities to supply .30 caliber weapons from U.S. stocks in the weapons containers dropped to Resistance groups sponsored by an SOE, or later also Office of Strategic Services (OSS), organizer, on the assumption the groups so supplied would be operating in areas within the operational boundaries of U.S. forces committed to Operation Overlord. They were found to be suited to the kind of operation the two British, two French, and one Belgian Regiment carried out. It was handy enough to parachute with, and, in addition, could be easily stowed in an operational Jeep. Other specialist intelligence collection units, such as 30 Assault Unit sponsored by the Naval Intelligence Division of the British Admiralty, which operated across the entire Allied area of operations, also made use of this weapon.. The carbine continued to be utilized as late as the Malayan Emergency, by the Police Field Force of the Royal Malaysian Police, along with other units of the British Army, were issued the M2 carbine for both jungle patrols and outpost defense. The Royal Ulster Constabulary also used the M1 carbine.


title: M1 carbine

text: Small numbers of captured M1 carbines were used by German forces in World War II, particularly after D-Day. The German designation for captured carbines was "Selbstladekarabiner 455(a)". The ""a"" came from the country name in German; in this case, "Amerika". It was also used by German police and border guards in Bavaria after World War II and into the 1950s. The carbines were stamped according to the branch they were in service with; for instance, those used by the border guard were stamped ""Bundesgrenzschutz"". Some of these weapons were modified with different sights, finishes, and sometimes new barrels.


title: M1 carbine

text: A variant was produced shortly after World War II by Japanese manufacturer Howa Machinery, under U.S. supervision. These were issued to all branches of the Japan Self-Defense Forces, and large numbers of them found their way to Southeast Asia during the Vietnam War. Howa also made replacement parts for US-made M1 carbines issued to Japanese police and military.


title: M1 carbine

text: The M1 carbine was also used by the Israeli Palmach-based special forces in the 1948 Arab–Israeli War. And, because of their compact size and semi-auto capabilities, they continued to be used by Israeli Defence Forces after the creation of Israel. The Israeli police still use the M1 carbine as a standard long gun for non-combat elements and Mash'az volunteers.


title: M1 carbine

text: The U.S. provided France with 269,644 M1 and M2 carbines from World War II to 1963. The carbines were used by the French Paratroopers and Legionnaires, as well as specialists (e.g., drivers, radio operators, engineers), during the Indo-China War, the Algerian War and the Suez Crisis.


title: M1 carbine

text: The U.S. provided the Army of the Republic of Vietnam with 793,994 M1 and M2 carbines from 1963 to 1973. Along with tens of thousands of carbines left behind by the French after the First Indochina War, the M1 and M2 carbines were the most widely issued small arm during the early stages in the Vietnam War and remained in service in large numbers until the fall of Saigon. The South Vietnamese would also receive 220,300 M1 Garands and 520 M1C/M1D rifles, and 640,000 M-16 rifles.


title: M1 carbine

text: The Viet Minh and the Viet Cong also used large numbers of M1 and M2 carbines, captured from the French, ARVN and local militia forces of South Vietnam, as well as receiving many thousands of carbines from the North Vietnamese Army (NVA), China and North Korea. Over time, the SKS and eventually the AK-47 would replace the carbine to become the dominant weapons used by the Viet Cong.


title: M1 carbine

text: The Republic of Korea Armed Forces received 1,015,568 M1 and M2 carbines from 1963 to 1972. Along with hundreds of thousands of Carbines and M1 Garands provided by the United States Army before, during and shortly after the Korean war, South Korea would become the largest single recipient of American M1 and M2 carbines.


title: M1 carbine

text: South Korea also took an active role in the Vietnam War. From 1964 to 1973, South Korea sent more than 300,000 troops to South Vietnam armed primarily with M1 and M2 carbines, as well as M1 Garands.


title: M1 carbine

text: The government of the Philippines still issues M1 carbines to the infantrymen of the Philippine Army's 2nd Infantry Division assigned in Luzon Island (some units are issued just M14 automatic rifles and M1 carbines) and the Civilian Auxiliary Forces Geographical Unit (CAFGU) and Civilian Volunteer Organizations (CVO) spread throughout the Philippines. Certain provincial police units of the Philippine National Police (PNP) still use government-issued M1 carbines as well as some operating units of the National Bureau of Investigation (NBI). In many provinces in the Philippines, M1 carbines are still a highly valued light small arm. Elements of the New People's Army and Islamic Secessionist movement value the carbine as a lightweight weapon and preferred choice for mountain and ambush operations.


title: M1 carbine

text: The M1 carbine has become one of the most recognized firearms in Philippine society, with the Marikina-based company ARMSCOR Philippines still continues to manufacture .30 caliber ammunition for the Philippine market.


title: M1 carbine

text: The M1 and M2 carbines were widely used by military, police, and security forces and their opponents during the many guerrilla and civil wars throughout Latin America until the 1990s, when they were mostly replaced by more modern designs. A notable user was Che Guevara who used them during the Cuban Revolution and in Bolivia where he was executed by a Bolivian soldier armed with an M2 carbine. Guevara's fellow revolutionary Camilo Cienfuegos also used an M2 carbine that he modified with the pistol grip and foregrip from a Thompson submachine gun. Cienfuegos' carbine is on display in the Museum of the Revolution (Cuba).


title: M1 carbine

text: In Rio de Janeiro, Brazil, a police battalion named "Batalhão de Operações Policiais Especiais" (BOPE, or "Special Police Operations Battalion") still uses the M1 carbine.


title: M1 carbine

text: The unit data provided below refers to original U.S. Ordnance contract carbines the United States provided these countries. Many countries sold, traded, destroyed, and/or donated these carbines to other countries and/or private gun brokers.


title: M1 carbine

text: The standard-issue versions of the carbine officially listed and supported were the M1, M1A1, M2 and M3.


title: M1 carbine

text: The M1A1 was designed in May 1942 for paratrooper units, and came with a folding stock, but was otherwise identical to a standard M1. M1A1 carbines were made by Inland, a division of General Motors and originally came with the early "L" nonadjustable sight and barrel band without bayonet lug. Inland production of M1A1 carbines was interspersed with Inland production of M1 carbines with the standard stock. Stocks were often swapped out as carbines were refurbished at arsenals.


title: M1 carbine

text: Initially, the M1 carbine was intended to have a selective-fire capability, but the decision was made to put the M1 into production without this feature. Fully automatic capability was incorporated into the design of the M2 (an improved, selective-fire version of the M1), introduced in 1944. The M2 featured the late M1 improvements to the rear sight, addition of a bayonet lug, and other minor changes.


title: M1 carbine

text: Research into a conversion kit for selective fire began May 1944; the first kit was developed by Inland engineers, and known as the "T4". Inland was awarded a contract for 500 T4 carbines in September 1944. Although the conversion was seen as satisfactory, the heavier 30-round magazine


title: M1 carbine

text: put greater strain on the magazine catch, necessitating the development of a sturdier catch. The slide, sear, and stock design also had to be modified. On fully automatic fire, the T4 model could fire about 750 rounds per minute, and generated a manageable recoil.


title: M1 carbine

text: Although some carbines were marked at the factory as M2, the only significant difference between an M1 and M2 carbine is in the fire control group. The military issued field conversion kits (T17 and T18) to convert an M1 to an M2. Legally a carbine marked M2 is always a machine gun for national firearms registry purposes.


title: M1 carbine

text: These M2 parts including the heavier M2 stock were standardized for arsenal rebuild of M1 and M1A1 carbines.


title: M1 carbine

text: A modified round bolt replaced the original flat top bolt to save machining steps in manufacture. Many sources erroneously refer to this round bolt as an "M2 bolt" but it was developed as a standard part for new manufacture M1 and later M2 carbines and as a replacement part, with priority given to use on M1A1 and M2 carbines. The slightly heavier round bolt did moderate the cyclic rate of the M2 on full automatic.


title: M1 carbine

text: Despite being in demand, very few M2 carbines saw use during World War II, and then mostly in the closing days against Japan. The M2 carbine was logistically compatible with the millions of M1 carbines in U.S. service, and offered longer range, better accuracy and better penetration than (pistol caliber) submachine guns like the M1 Thompsons and M3 Grease Guns. Therefore, after World War II, the M2 carbine largely replaced the submachine-guns in U.S. service, until it was itself replaced by the M16 rifle.


title: M1 carbine

text: The M2 model was the most widely used carbine variant during the Korean War. A detailed study of the effectiveness of the M2 in the war was assembled by S.L.A. Marshall. He found that many troops complained on the lack of effective range of the gun, which allowed the enemy to get close enough to throw hand grenades. A more detailed analysis showed however that most troops who complained actually tended to run low on ammo, because they fired their M2 on fully automatic too soon. Troops who fired their guns on semi-automatic at distance generally complained less about the M2's effectiveness. Generally, the more seasoned troops used the latter approach. The carbine was usually given to second line troops (administrative, support, etc.), who had little combat experience and also did not have much training in small-unit tactics, but who usually had to engage the enemy at some critical moment, like a breakthrough or ambush. Marshall noted that almost all killing shots with carbines in Korea were at ranges of 50 yards or less. The M2 was a preferred weapon for night patrols. The M2 was also used in the early stages of the Vietnam War by special forces, ARVN advisers, and air crews.


title: M1 carbine

text: Contemporary authors have struggled to categorize the M2 carbine. While it did introduce select-fire capability and an intermediate cartridge, its stopping power and weight was far below that of the StG 44. As such, it is considered by some to be an early predecessor to the assault rifle or personal defense weapon.


title: M1 carbine

text: The M3 carbine was an M2 carbine fitted with a mount designed to accept an infrared sight for use at night. It was initially used with the M1 sniperscope, and an active infrared sight, and saw action in 1945 with the Army during the invasion of Okinawa. Before the M3 carbine and M1 sniperscope were type-classified, they were known as the T3 and T120, respectively. The system continued to be developed, and by the time of the Korean War, the M3 carbine was used with the M3 sniperscope.


title: M1 carbine

text: The M2 sniper scope extended the effective nighttime range of the M3 carbine to 100 yards. In the later stages of the Korean War, an improved version of the M3 carbine, with a revised mount, a forward pistol grip, and a new M3 sniperscope design was used in the latter stages of Korea and briefly in Vietnam. The M3 sniperscope had a large active infrared spotlight mounted on top of the scope body itself, allowing use in the prone position. The revised M3/M3 had an effective range of around 125 yards. Eventually, the M3 carbine and its M3 sniperscope would be superseded by passive-design night vision scopes with extended visible ranges; the improved scopes in turn required the use of rifle-caliber weapons with flatter trajectories and increased hit probability.


title: M1 carbine

text: The Ingram SAM rifles are M1 carbine derivatives in 5.56×45mm NATO (SAM-1), 7.62×39mm (SAM-2) and 7.62×51mm NATO (SAM-3). The 5.56×45mm versions accept M16 magazines, the 7.62×39mm accept AK magazines and the 7.62×51mm versions use FN FAL magazines. They did not catch on in competition against the Ruger Mini-14 in both the police and civilian markets. The Ingram SAM rifles are occasionally found on auction sites for collectors.


title: M1 carbine

text: Iver Johnson's 9×19mm Parabellum carbine was introduced in 1985 until 1986 using modified Browning High Power 20-round magazines.


title: M1 carbine

text: Chiappa Firearms produces a 9mm M1 carbine derivative called the M1-9 which uses Beretta M9/92FS magazines. The Chiappa is not gas operated and instead relies on blowback operation.


title: M1 carbine

text: Several companies manufactured copies of the M1 carbine after World War II, which varied in quality. Some companies used a combination of original USGI and new commercial parts, while others manufactured entire firearms from new parts, which may or may not be of the same quality as the originals. These copies were marketed to the general public and police agencies but were not made for or used by the U.S. military.


title: M1 carbine

text: In 1963, firearms designer Col. Melvin M. Johnson developer of the M1941 Johnson rifle offered a conversion of US Military M1 carbines to his new Johnson MMJ 5.7mm Spitfire cartridge, while also introducing a newly manufactured version of the M1 carbine called the "Spitfire" made by his Johnson Arms, Inc. business that was designed and built specifically for this new 5.7 mm (.22 in) wildcat cartridge (also known as the 5.7 mm MMJ or .22 Spitfire). The Spitfire was advertised firing a 40-grain (2.6 g) bullet with a muzzle velocity of 3050 ft/s (930 m/s), though handloaders with careful selection of modern powders and appropriate bullets consistently safely exceed those numbers while remaining within the M1 carbine's maximum pressure rating of 38,500 psi (265 MPa). In comparison, the "standard" load for the .30 Carbine has a .30 Carbine ball bullet weighing 110 grains (7.1 g); a complete loaded round weighs 195 grains (12.6 g) and has a muzzle velocity of 1,990 ft/s (610 m/s), giving it 967 ft⋅lbf (1,311 joules) of energy when fired from the M1 carbine's 18-inch barrel.


title: M1 carbine

text: Johnson advertised the smaller caliber and the modified carbine as a survival rifle for use in jungles or other remote areas. It provided for light, easily carried ammunition in a light, fast handling carbine with negligible recoil. While the concept had some military application when used for this role in the selective-fire M2 carbine, it was not pursued, and few Spitfire carbines were made.


title: M1 carbine

text: More recently, the Auto-Ordnance division of Kahr Arms began production of an M1 carbine replica in 2005 based on the typical M1 carbine as issued in 1944, without the later adjustable sight or barrel band with bayonet lug. The original Auto-Ordnance had produced various parts for IBM carbine production during World War II, but did not manufacture complete carbines until the introduction of this replica. The AOM110 and AOM120 models (no longer produced) featured birch stocks and handguards, Parkerized receivers, flip-style rear sights and barrel bands without bayonet lugs. The current AOM130 and AOM140 models are identical except for American walnut stocks and handguards.


title: M1 carbine

text: In 2014, Inland Manufacturing, LLC in Dayton, Ohio introduced the reproduction of the "Inland M1 carbine". Inland Manufacturing, LLC is a private entity that is producing reproductions of the M1 carbine and M1A1 paratrooper models that were built by the original Inland Division of General Motors from 1941 to 1945. The new Inland M1 carbines feature many of the same characteristics of the original Inland carbines and are manufactured in the US. The M1 carbine is modeled after the last production model that Inland manufactured in 1945 and features a Type 3 bayonet lug and barrel band, adjustable rear sights, push button safety, round bolt, and "low wood" walnut stock, and a 15-round magazine. A 30-round mag catch was utilized to allow high-capacity magazines. A "1944" M1 carbine is also available that has the same features as the 1945 only with a Type 2 barrel Band and 10-round magazine and is available for sale in most states with magazine capacity and bayonet lug restrictions. The M1A1 is modeled after a late production 1944 M1A1 paratrooper model with a folding "low wood" walnut stock, Type two barrel band, and includes the same adjustable sights which were actually introduced in 1944.


title: M1 carbine

text: An Israeli arms company (Advanced Combat Systems) offers a modernized bullpup variant called the Hezi SM-1. The company claims accuracy of 1.5 MOA at .


title: M1 carbine

text: Some U.S. states prohibit use of the .30 Carbine cartridge for hunting deer and larger animals due to a lessened chance of killing an animal in a single shot, even with expanding bullets. The M1 carbine is also prohibited for hunting in several states such as Pennsylvania because of the semi-automatic function, and Illinois which prohibits all non-muzzleloading rifles for big game hunting. Five-round magazines are commercially made for use in states that limit the capacity of semi-automatic hunting rifles.


title: M1 carbine

text: The M1 carbine was also used by various law enforcement agencies and prison guards, and was prominently carried by riot police during the civil unrest of the late 1960s and early 1970s; until it was replaced in those roles by more modern .223 caliber semi-automatic rifles such as the Ruger Mini-14 and the Colt AR-15-type rifles in the late 1970s and early 1980s.


title: M1 carbine

text: The ease of use and great adaptability of the weapon led to it being used by Malcolm X and Patty Hearst. Both were featured in famous news photographs carrying a version the carbine. One of these firearms was also the weapon used in the 1947 assassination of notorious American mobster Benjamin "Bugsy" Siegel.


title: Pussy

text: Pussy is a used as a noun, an adjective, and—in rare instances—a verb in the English language. It has several meanings, as slang, as euphemism, and as vulgarity. The most common as a noun, it means "cat", as well as "coward or weakling". In slang usage, it can mean "the human vulva or vagina" and less commonly, as a form of synecdoche, meaning "sexual intercourse with a woman". Because of its multiple senses including both innocent and vulgar connotations, "pussy" is often the subject of double entendre.


title: Pussy

text: The etymology of the word is not clear. Several different senses of the word have different histories or origins. The earliest records of "pussy" are in the 19th century, meaning something fluffy.


title: Pussy

text: The noun "pussy" meaning "cat" comes from the Modern English word "puss", a conventional name or term of address for a pet cat. The "Oxford English Dictionary" (OED) says that cognates are common to several Germanic languages, including Dutch "poes" and Middle Low German "pūse", which are also used to call a cat. The word "puss" is attested in English as early as 1533. Earlier etymology is uncertain, but similar words exist in other European languages, including Lithuanian "puižė" and Irish "puisín", both traditional calls to attract a cat.


title: Pussy

text: The words "puss" and derived forms "pussy" and "pusscat" were extended to refer to girls or women by the seventeenth century. This sense of "pussy" was used to refer specifically to genitalia by the eighteenth century, and from there further extended to refer to sexual intercourse involving a woman by the twentieth century.


title: Pussy

text: Noah Webster, in his original 1828 "American Dictionary of the English Language", defined "pussy" as: "inflated, swelled; hence, fat, short and thick; and as persons of this make labor in respiration, the word is used for short breathed". He gave "pursy" as a "corrupt orthography" or misspelling of "pussy". In 1913, however, "Webster's Revised Unabridged Dictionary" reversed the original, suggesting that "pussy" was a "colloquial or low" variant of "pursy". That word, in turn, was defined as "fat and short-breathed", with etymology from Old French "pousser" "to push". In the Northern English variant of British English, the dialect form "pursy", now rare or obsolete, means "fat" or "short-winded".


title: Pussy

text: The "Webster's Third International Dictionary" points out similarities between "pussy" in the sense of "vulva" and Low German or Scandinavian words meaning "pocket" or "purse", including Old Norse "pūss" and Old English "pusa".


title: Pussy

text: The medieval French word "pucelle", meaning "maiden" or "virgin", is not related to the English word. It is attested in Old French from the ninth century, and likely derives from Latin. The precise Latin source is disputed, with either "puella" "girl" or "pulla" "pullet, young female chicken" suggested as earlier sources.


title: Pussy

text: As a homograph, "pussy" also has the meaning "containing pus"; with this meaning, the word is pronounced , while the other forms are all pronounced .


title: Pussy

text: Meanings of the verb relate to the common noun senses, including "to act like a cat", "to act like a coward", or "to have sex with a woman". Adjective meanings are likewise related to the noun.


title: Pussy

text: Both in English and in German "puss" was used as a "call-name" for cats, but in English "pussy" was used as a synonym for the word "cat" in other uses as well. In addition to cats, the word was also used for rabbits and hares. In the 19th century, the meaning was extended to anything soft and furry. "Pussy willow", for example, is a name applied to various species in the genus "Salix" with furry catkins. In thieves' cant the word "pussy" means a "fur coat".


title: Pussy

text: The "Oxford English Dictionary" gives as the first meaning of the noun: "Chiefly colloq[uial]. A girl or woman exhibiting characteristics associated with a cat, esp[ecially] sweetness or amiability. Freq[uently] used as a pet name or as a term of endearment." The examples it cites from the mid-19th to mid-20th centuries are not sexual. Another example, not cited by the OED, is one of the main characters of E. Nesbit's "Five Children and It" - Jane, nicknamed Pussy by her siblings.


title: Pussy

text: The verb "pussyfoot", meaning to walk softly or to speak in an evasive or cautious manner, may come from the adjective "pussy-footed" "having a cat-like foot", or directly from the noun "pussyfoot". This word, first attested in the late nineteenth century, is related to both the "cat" and the "woman" meanings of "pussy".


title: Pussy

text: In contemporary English, use of the word "pussy" to refer to women themselves is considered derogatory and demeaning, treating people as sexual objects. As a reference to genitals or to sexual intercourse, the word is considered vulgar slang. Studies find the word is used more commonly in conversations among men than in groups of women or mixed-gender groups, though subjects report using "pussy" more often than other slang terms for female genitals. There are women seeking to reclaim the word to symbolise sexual pleasure, power, and trust in their bodies (e.g. around childbirth).


title: Pussy

text: Donald Trump's use of the word to describe celebrity interactions with women ("grab them by the pussy", known as the "Access Hollywood" tape) provoked strong reactions by media figures and politicians across the political spectrum; an image of a snarling cat with the slogan "pussy grabs back" became a "rallying cry for female rage against Trump". Pink "pussyhats" (knitted caps with cat-like ears) were a notable feature of the worldwide protests held the day after Trump's inauguration as President of the United States. The name attempts to reclaim the derogatory term and was never intended to be an anatomical representation.


title: Pussy

text: Words referring to cats are used as vulgar slang for female genitals in some other European languages as well. Examples include German "Muschi" (literally "house cat"), French "chatte" ("female cat", also used to refer to sexual intercourse), and Dutch "poes" ("puss"). The Portuguese term "rata" (literally "female rat") and Norwegian "mus" ("mouse") are also animal terms used as vulgar slang for women's genitals.


title: Pussy

text: The word "pussy" is also used in a derogatory sense to mean cowardly, weak, or easily fatigued. The "Collins Dictionary" says: taboo, slang, mainly US) an ineffectual or timid person." It may refer to a male who is not considered sufficiently masculine, as in: "The coach calls us pussies."


title: Pussy

text: Men who are dominated by women (particularly by their partners or spouses and at one time referred to as "hen-pecked"; see pecking order) can be referred to as "pussy-whipped" (or simply "whipped" in slightly more polite society or media). This may be used simply to denigrate a man who is contented in a relationship. The hyphenated phrase is parsed as "whipped by pussy", a manipulative relationship dynamic wherein a female deliberately or subconsciously withholds sexual intercourse to coerce the male into surrendering power in other aspects of the relationship. The male's weakness is his desire for access to female genitalia, and his willingness to weaken his position in the relationship to obtain that access, combining two uses of the word pussy.


title: Pussy

text: Pussy is one of a large number of English words that has both erotic and non-erotic meanings. Such "double entendres" have long been used in the creation of sexual humor. This double meaning of "pussy" has been used for over a hundred years by performers, including the late-19th-century vaudeville act the Barrison Sisters, who performed the notorious routine "Do You Want To See My Pussy?" in which they raised their skirts to reveal live kittens.


title: Pussy

text: In the British comedy "Are You Being Served?" the character Mrs. Slocombe often expressed concern for the welfare of her pussy. The "double entendre" made every reference to her cat seem to be a salacious and therefore humorous reference to her vulva.


title: Pussy

text: In the 2002 film "8 Mile", a rapper insults his rivals by including the line, "How can six dicks be pussies?" The line relies on double meanings of both "dick" (either "contemptible person" or "male genitalia") and "pussy" ("weak" or "female genitalia"). Such word play presents a challenge for translators of the film.


title: Pussy

text: Pussy Riot is a Russian radical feminist punk rock collective that stages illegal events in Moscow protesting President Vladimir Putin and the status of women in Russian society. Band member "Kot" says that she knows how the word is used in English, and that it is also used in Russian as term of endearment for little girls. These various meanings create a tension with the word "riot", which the group likes.


title: Pussy

text: In 2017 Planned Parenthood released a series of short videos on YouTube about female sexual health, with the overall title "How to take care of your pussy". Instead of the word "pussy" being shown or spoken, a cat appears instead. The visuals consist mainly of cats, playing on the popularity of cat videos, with a voiceover by Sasheer Zamata. "Refinery29" called it "a pretty genius metaphor" and "Metro" said: "If there are two things left in this world that are inherently wonderful, it’s cats and vaginas. Don’t argue. It’s true.[...] It makes sense, then, that Planned Parenthood has decided to combine the two to create a truly splendid video series." The series has been shortlisted for a Shorty Award.


title: Alexithymia

text: Alexithymia is a personality trait characterized by the inability to identify and describe emotions experienced by oneself. The core characteristic of alexithymia is marked dysfunction in emotional awareness, social attachment, and interpersonal relation. Furthermore, people with high levels of alexithymia can have difficulty distinguishing and appreciating the emotions of others, which is thought to lead to unempathic and ineffective emotional responses.


title: Alexithymia

text: High levels of alexithymia occur in approximately 10% of the population and can occur with a number of psychiatric conditions as well as any neurodevelopmental disorder. Difficulty with recognizing and talking about their emotions appears at subclinical levels in men who conform to cultural notions of masculinity (such as thinking that sadness is a feminine emotion). This is called normative male alexithymia by some researchers. However, both alexithymia itself and its association with traditionally masculine norms occur in both sexes.


title: Alexithymia

text: Alexithymia is considered to be a personality trait that places affected individuals at risk for other medical and psychiatric disorders while reducing the likelihood that these individuals will respond to conventional treatments for the other conditions. Alexithymia is not classified as a mental disorder in the fifth edition of the "Diagnostic and Statistical Manual of Mental Disorders". It is a dimensional personality trait that varies in intensity from person to person. A person's alexithymia score can be measured with questionnaires such as the Toronto Alexithymia Scale (TAS-20), the Perth Alexithymia Questionnaire (PAQ), the Bermond-Vorst Alexithymia Questionnaire (BVAQ), the Levels of Emotional Awareness Scale (LEAS), the Online Alexithymia Questionnaire (OAQ-G2), the Toronto Structured Interview for Alexithymia (TSIA), or the Observer Alexithymia Scale (OAS). It is distinct from the psychiatric personality disorders, such as antisocial personality disorder.


title: Alexithymia

text: However, there is some ongoing disagreement in the field about the definition of alexithymia. When measured in empirical studies, constricted imaginal processes are often found not to statistically cohere with the other components of alexithymia. Such findings have led to debate in the field about whether IMP is indeed a component of alexithymia. For example, in 2017, Preece and colleagues introduced the attention-appraisal model of alexithymia, where they suggested that IMP be removed from the definition and that alexithymia be conceptually composed only of DIF, DDF, and EOT, as each of these three are specific to deficits in emotion processing. In practice, since the constricted imaginal processes items were removed from earlier versions of the TAS-20 in the 1990s, the most used alexithymia assessment tools (and consequently most alexithymia research studies) have only assessed the construct in terms of DIF, DDF, and EOT.


title: Alexithymia

text: Studies (using measures of alexithymia assessing DIF, DDF, and EOT) have reported that the prevalence rate of high alexithymia is less than 10% of the population. A less common finding suggests that there may be a higher prevalence of alexithymia amongst males than females, which may be accounted for by difficulties some males have with "describing feelings", but not by difficulties in "identifying feelings" in which males and females show similar abilities.


title: Alexithymia

text: Psychologist R. Michael Bagby and psychiatrist Graeme J. Taylor have argued that the alexithymia construct is inversely related to the concepts of psychological mindedness and emotional intelligence and there is "strong empirical support for alexithymia being a stable personality trait rather than just a consequence of psychological distress".


title: Alexithymia

text: Typical deficiencies may include problems identifying, processing, describing, and working with one's own feelings, often marked by a lack of understanding of the feelings of others; difficulty distinguishing between feelings and the bodily sensations of emotional arousal; confusion of physical sensations often associated with emotions; few dreams or fantasies due to restricted imagination; and concrete, realistic, logical thinking, often to the exclusion of emotional responses to problems. Those who have alexithymia also report very logical and realistic dreams, such as going to the store or eating a meal. Clinical experience suggests it is the structural features of dreams more than the ability to recall them that best characterizes alexithymia.


title: Alexithymia

text: Some alexithymic individuals may appear to contradict the above-mentioned characteristics because they can experience chronic dysphoria or manifest outbursts of crying or rage. However, questioning usually reveals that they are quite incapable of describing their feelings or appear confused by questions inquiring about specifics of feelings.


title: Alexithymia

text: According to Henry Krystal, individuals exhibiting alexithymia think in an operative way and may appear to be superadjusted to reality. In psychotherapy, however, a cognitive disturbance becomes apparent as patients tend to recount trivial, chronologically ordered actions, reactions, and events of daily life with monotonous detail. In general, these individuals can, but not always, seem oriented toward things and even treat themselves as robots. These problems seriously limit their responsiveness to psychoanalytic psychotherapy; psychosomatic illness or substance abuse is frequently exacerbated should these individuals enter psychotherapy.


title: Alexithymia

text: A common misconception about alexithymia is that affected individuals are totally unable to express emotions verbally and that they may even fail to acknowledge that they experience emotions. Even before coining the term, Sifneos (1967) noted patients often mentioned things like anxiety or depression. The distinguishing factor was their inability to elaborate beyond a few limited adjectives such as "happy" or "unhappy" when describing these feelings. The core issue is that people with alexithymia have poorly differentiated emotions limiting their ability to distinguish and describe them to others. This contributes to the sense of emotional detachment from themselves and difficulty connecting with others, making alexithymia negatively associated with life satisfaction even when depression and other confounding factors are controlled for.


title: Alexithymia

text: Alexithymia frequently co-occurs with other disorders. Research indicates that alexithymia overlaps with autism spectrum disorders (ASD). In a 2004 study using the TAS-20, 85% of the adults with ASD fell into the "impaired" category and almost half fell into the "severely impaired" category; in contrast, among the adult control population only 17% were "impaired", none "severely impaired". Fitzgerald & Bellgrove pointed out that, "Like alexithymia, Asperger's syndrome is also characterised by core disturbances in speech and language and social relationships". Hill & Berthoz agreed with Fitzgerald & Bellgrove (2006) and in response stated that "there is some form of overlap between alexithymia and ASDs". They also pointed to studies that revealed impaired theory of mind skill in alexithymia, neuroanatomical evidence pointing to a shared etiology and similar social skills deficits. The exact nature of the overlap is uncertain. Alexithymic traits in AS may be linked to clinical depression or anxiety; the mediating factors are unknown and it is possible that alexithymia predisposes to anxiety. On the other hand, while the total alexithymia score as well as the difficulty in identifying feelings and externally oriented thinking factors are found to be significantly associated with ADHD, and while the total alexithymia score, the difficulty in identifying feelings, and the difficulty in describing feelings factors are also significantly associated with symptoms of hyperactivity and impulsivity, there is no significant relationship between alexithymia and inattentiveness symptom.


title: Alexithymia

text: There are many more psychiatric disorders that overlap with alexithymia. One study found that 41% of US veterans of the Vietnam War with post-traumatic stress disorder (PTSD) were alexithymic. Another study found higher levels of alexithymia among Holocaust survivors with PTSD compared to those without. Higher levels of alexithymia among mothers with interpersonal violence-related PTSD were found in one study to have proportionally less caregiving sensitivity. This latter study suggested that when treating adult PTSD patients who are parents, alexithymia should be assessed and addressed also with attention to the parent-child relationship and the child's social-emotional development.


title: Alexithymia

text: Single study prevalence findings for other disorders include 63% in anorexia nervosa, 56% in bulimia, 45% to 50% in major depressive disorder, 34% in panic disorder, 28% in social phobia, and 50% in substance abusers. Alexithymia is also exhibited by a large proportion of individuals with acquired brain injuries such as stroke or traumatic brain injury.


title: Alexithymia

text: Alexithymia is correlated with certain personality disorders, particularly schizoid, avoidant, dependent and schizotypal, substance use disorders, some anxiety disorders and sexual disorders as well as certain physical illnesses, such as hypertension, inflammatory bowel disease, diabetes and functional dyspepsia. Alexithymia is further linked with disorders such as migraine headaches, lower back pain, irritable bowel syndrome, asthma, nausea, allergies and fibromyalgia.


title: Alexithymia

text: An inability to modulate emotions is a possibility in explaining why some people with alexithymia are prone to discharge tension arising from unpleasant emotional states through impulsive acts or compulsive behaviors such as binge eating, substance abuse, perverse sexual behavior or anorexia nervosa. The failure to regulate emotions cognitively might result in prolonged elevations of the autonomic nervous system (ANS) and neuroendocrine systems, which can lead to somatic diseases. People with alexithymia also show a limited ability to experience positive emotions leading Krystal and Sifneos (1987) to describe many of these individuals as anhedonic.


title: Alexithymia

text: Alexisomia is a clinical concept that refers to the difficulty in the awareness and expression of somatic, or bodily, sensations. The concept was first proposed in 1979 by Dr. Yujiro Ikemi when he observed characteristics of both alexithymia and alexisomia in patients with psychosomatic diseases.


title: Alexithymia

text: Early studies showed evidence that there may be an interhemispheric transfer deficit among people with alexithymia; that is, the emotional information from the right hemisphere of the brain is not being properly transferred to the language regions in the left hemisphere, as can be caused by a decreased corpus callosum, often present in psychiatric patients who have suffered severe childhood abuse. A neuropsychological study in 1997 indicated that alexithymia may be due to a disturbance to the right hemisphere of the brain, which is largely responsible for processing emotions. In addition, another neuropsychological model suggests that alexithymia may be related to a dysfunction of the anterior cingulate cortex. These studies have some shortcomings, however, and the empirical evidence about the neural mechanisms behind alexithymia remains inconclusive.


title: Alexithymia

text: French psychoanalyst Joyce McDougall objected to the strong focus by clinicians on neurophysiological explanations at the expense of psychological ones for the genesis and operation of alexithymia, and introduced the alternative term "disaffectation" to stand for psychogenic alexithymia. For McDougall, the disaffected individual had at some point "experienced overwhelming emotion that threatened to attack their sense of integrity and identity", to which they applied psychological defenses to pulverize and eject all emotional representations from consciousness. A similar line of interpretation has been taken up using the methods of phenomenology. McDougall has also noted that all infants are born unable to identify, organize, and speak about their emotional experiences (the word "infans" is from the Latin "not speaking"), and are "by reason of their immaturity inevitably alexithymic". Based on this fact McDougall proposed in 1985 that the alexithymic part of an adult personality could be "an extremely arrested and infantile psychic structure". The first language of an infant is nonverbal facial expressions. The parent's emotional state is important for determining how any child might develop. Neglect or indifference to varying changes in a child's facial expressions without proper feedback can promote an invalidation of the facial expressions manifested by the child. The parent's ability to reflect self-awareness to the child is another important factor. If the adult is incapable of recognizing and distinguishing emotional expressions in the child, it can influence the child's capacity to understand emotional expressions.


title: Alexithymia

text: Molecular genetic research into alexithymia remains minimal, but promising candidates have been identified from studies examining connections between certain genes and alexithymia among those with psychiatric conditions as well as the general population. A study recruiting a test population of Japanese males found higher scores on the Toronto Alexithymia Scale among those with the 5-HTTLPR homozygous long (L) allele. The 5-HTTLPR region on the serotonin transporter gene influences the transcription of the serotonin transporter that removes serotonin from the synaptic cleft, and is well studied for its association with numerous psychiatric disorders. Another study examining the 5-HT1A receptor, a receptor that binds serotonin, found higher levels of alexithymia among those with the G allele of the Rs6295 polymorphism within the HTR1A gene. Also, a study examining alexithymia in subjects with obsessive-compulsive disorder found higher alexithymia levels associated with the Val/Val allele of the Rs4680 polymorphism in the gene that encodes Catechol-O-methyltransferase (COMT), an enzyme which degrades catecholamine neurotransmitters such as dopamine. These links are tentative, and further research will be needed to clarify how these genes relate to the neurological anomalies found in the brains of people with alexithymia.


title: Alexithymia

text: Although there is evidence for the role of environmental and neurological factors, the role and influence of genetic factors for developing alexithymia is still unclear. A single large scale Danish study suggested that genetic factors contributed noticeably to the development of alexithymia. However, some scholars find twin studies and the entire field of behavior genetics to be controversial. Those scholars raise concerns about the "equal environments assumption". Traumatic brain injury is also implicated in the development of alexithymia, and those with traumatic brain injury are six times more likely to exhibit alexithymia.


title: Alexithymia

text: Alexithymia can create interpersonal problems because these individuals tend to avoid emotionally close relationships, or if they do form relationships with others they usually position themselves as either dependent, dominant, or impersonal, "such that the relationship remains superficial". Inadequate "differentiation" between self and others by alexithymic individuals has also been observed. Their difficulty in processing interpersonal connections often develops where the person lacks a romantic partner.


title: Alexithymia

text: In a study, a large group of alexithymic individuals completed the 64-item Inventory of Interpersonal Problems (IIP-64) which found that "two interpersonal problems are significantly and stably related to alexithymia: cold/distant and non-assertive social functioning. All other IIP-64 subscales were not significantly related to alexithymia."


title: Alexithymia

text: Chaotic interpersonal relations have also been observed by Sifneos. Due to the inherent difficulties identifying and describing emotional states in self and others, alexithymia also negatively affects relationship satisfaction between couples.


title: Alexithymia

text: In a 2008 study alexithymia was found to be correlated with impaired understanding and demonstration of relational affection, and that this impairment contributes to poorer mental health, poorer relational well-being, and lowered relationship quality. Individuals high on the alexithymia spectrum also report less distress at seeing others in pain and behave less altruistically toward others.


title: Alexithymia

text: Some individuals working for organizations in which control of emotions is the norm might show alexithymic-like behavior but not be alexithymic. However, over time the lack of self-expressions can become routine and they may find it harder to identify with others.


title: Alexithymia

text: Generally speaking, approaches to treating alexithymia are still in their infancy, with not many proven treatment options available.


title: Alexithymia

text: In 2002, Kennedy and Franklin found that a skills-based intervention is an effective method for treating alexithymia. Kennedy and Franklin's treatment plan involved giving the participants a series of questionnaires, psychodynamic therapies, cognitive-behavioral and skills-based therapies, and experiential therapies. After treatment, they found that participants were generally less ambivalent about expressing their emotion feelings and more attentive to their emotional states.


title: Alexithymia

text: In 2017, based on their attention-appraisal model of alexithymia, Preece and colleagues recommended that alexithymia treatment should target trying to improve the developmental level of people's emotion schemas and reduce people's use of experiential avoidance of emotions as an emotion regulation strategy (i.e., the mechanisms hypothesized to underlie alexithymia difficulties in the attention-appraisal model of alexithymia).


title: Alexithymia

text: In 2018, Löf, Clinton, Kaldo, and Rydén found that mentalisation-based treatment is also an effective method for treating alexithymia. Mentalisation is the ability to understand the mental state of oneself or others that underlies overt behavior, and mentalisation-based treatment helps patients separate their own thoughts and feelings from those around them. This treatment is relational, and it focuses on gaining a better understanding and use of mentalising skills. The researchers found that all of the patients' symptoms including alexithymia significantly improved, and the treatment promoted affect tolerance and the ability to think flexibly while expressing intense affect rather than impulsive behavior.


title: Alexithymia

text: A significant issue impacting alexithymia treatment is that alexithymia has comorbidity with other disorders. Mendelson's 1982 study showed that alexithymia frequently presented in people with undiagnosed chronic pain. Participants in Kennedy and Franklin's study all had anxiety disorders in conjunction with alexithymia, while those in Löf et al. were diagnosed with both alexithymia and borderline personality disorder. All these comorbidity issues complicate treatment because it is difficult to examine people who exclusively have alexithymia.


title: Alexithymia

text: The term "alexithymia" was coined by psychotherapists John Case Nemiah and Peter Sifneos in 1973. The word comes from Greek: (, 'not', privative prefix, alpha privative) + (, 'words') + (, 'disposition', 'feeling', or 'rage') (cf. dyslexia), literally meaning "no words for emotions".


title: Alexithymia

text: Nonmedical terms describing similar conditions include emotionless and impassive. People with the condition are called alexithymics or alexithymiacs.


title: .45 ACP

text: The .45 ACP (Automatic Colt Pistol) or .45 Auto (11.43×23mm) is a rimless straight-walled handgun cartridge designed by John Moses Browning in 1904, for use in his prototype Colt semi-automatic pistol. After successful military trials, it was adopted as the standard chambering for Colt's M1911 pistol. The round was developed due to a lack of stopping power experienced in the Moro Rebellion in places like Sulu. The issued ammunition, .38 Long Colt, had proved inadequate, motivating the search for a better cartridge. This experience and the Thompson–LaGarde Tests of 1904 led the Army and the Cavalry to decide that a minimum of .45 caliber was required in a new handgun.


title: .45 ACP

text: The standard issue military .45 ACP round uses a 230-grain (14.9 g) round nose projectile that travels at approximately 830 feet per second (250 m/s) when fired from a government-issue M1911A1 pistol. It operates at a relatively low maximum chamber pressure rating of , compared to for both 9mm Parabellum and .40 S&W, which due to a low bolt thrust helps extend the service lives of weapons. Since standard-pressure .45 ACP rounds are subsonic when fired from handguns and submachine guns, it is a useful caliber for suppressed weapons to eliminate the sonic boom.


title: .45 ACP

text: Today, most NATO militaries use sidearms chambered for the 9×19mm Parabellum cartridge, but the effectiveness of the .45 ACP cartridge has ensured its continued popularity with large-caliber sport shooters, especially in the United States. In 1985, the .45 ACP M1911A1 pistol was replaced by the Beretta M9 9 mm pistol as the main sidearm of the U.S. military, which in turn was replaced with the SIG Sauer P320 9 mm pistol in 2017, designated M17 for the full-size and M18 for the compact.


title: .45 ACP

text: During the late 19th century and early 20th centuries, the U.S. Cavalry began trials to replace their sidearm arsenal of issued .45 Colt Single Action Army (SAA) in favor of the more modern and versatile double-action revolver in .45 Colt.


title: .45 ACP

text: After the example of the Cavalry, the Army in turn had fielded versions of double-action revolvers in .38 Long Colt. It was eventually evaluated that the .38 caliber round was significantly less effective in overall stopping power than the .45 Colt against determined opponents in cases such as the Tausug Moro juramentado warriors, who were encountered in the Moro Rebellion. The standard-issue rifle, the .30-40 Krag, had also failed to stop Moro warriors effectively; the British had similar lack-of-stopping-power issues switching to the .303 British, which resulted in the development of the "dum-dum" bullet, in an attempt to compensate for the round's deficiencies. This experience, and the Thompson–LaGarde Tests of 1904, led the Army and the Cavalry to decide a minimum of .45 caliber was required in a new handgun. Thompson and Major Louis Anatole La Garde of the medical corps arranged tests on cadavers and animals in the Chicago stockyards, resulting in their declaring that the .45 was the most effective pistol cartridge. They noted, however, training was critical to make sure a soldier could score a hit in a vulnerable part of the body.


title: .45 ACP

text: Colt had been working with Browning on a .41 caliber cartridge in 1904, and in 1905, when the Cavalry asked for a .45-caliber equivalent, Colt modified the pistol design to fire an enlarged version of the prototype .41 round. The result from Colt was the Model 1905 and the new .45 ACP cartridge. The original round that passed the testing fired a bullet at , but after a number of rounds of revisions between Winchester Repeating Arms, Frankford Arsenal, and Union Metallic Cartridge, it ended up using a bullet fired at a nominal velocity of . The resulting .45 caliber cartridge, named the ".45 ACP", was similar in performance to the .45 Schofield cartridge and only slightly less powerful while significantly shorter than the .45 Colt cartridge that the United States Cavalry was using at the time.


title: .45 ACP

text: By 1906, bids from six makers were submitted, among them, Browning's design, submitted by Colt. Only DWM, Savage, and Colt made the first cut. DWM, which submitted two Parabellums chambered in .45 ACP, withdrew from testing after the first round of tests, for unspecified reasons.


title: .45 ACP

text: In the second round of evaluations in 1910, the Colt design passed extensive testing with no failures, while the Savage design suffered 37 stoppages or parts failures. The Colt pistol was adopted as the Model 1911.


title: .45 ACP

text: The cartridge-pistol combination was quite successful but not satisfactory for U.S. military purposes. Over time, a series of improved designs were offered, culminating in the adoption in 1911 of the "Cal. 45 Automatic Pistol Ball Cartridge, Model of 1911", a round with a bullet weight of . The first production, at Frankford Arsenal, was marked "F A 8 11", for the August 1911 date.


title: .45 ACP

text: The cartridge was designed by John Browning for Colt, but the most influential person in selecting the cartridge was Army ordnance member General John T. Thompson. After the poor performance of the Army's .38 Long Colt pistols evidenced during the Philippine–American War (1899–1902), Thompson insisted on a more capable pistol cartridge.


title: .45 ACP

text: The "T" (trials) designation was used for the experimental version of a cartridge and "M" (model) is used for the accepted and standardized version.


title: .45 ACP

text: It came in either 24-round revolver ammo cartons, containing eight 3-round "half-moon" clips (1917-1945), pre-war 20-round cartons (1911-1942), or wartime 50-round cartons (1942–present). The M12 and M15 shotshell rounds were packed in 20-round cartons sealed in foil-lined paper.


title: .45 ACP

text: "S.A." stands for small arms. The "z" in the designation stands for cartridges loaded with nitrocellulose rather than cordite.


title: .45 ACP

text: The common rifling twist rate for this cartridge is 1 in , 6 grooves, Ø lands = .442 in (11.23 mm), Ø grooves = 45 in (11.43 mm), land width = .147 in (3.73 mm) and the primer type is large pistol.


title: .45 ACP

text: According to Commission internationale permanente pour l'épreuve des armes à feu portatives (CIP) rulings, the .45 ACP cartridge case can handle up to P piezo pressure. In CIP-regulated countries every pistol cartridge combination has to be proofed at 130% of this maximum CIP pressure to certify for sale to consumers. This means that .45 ACP chambered arms in CIP-regulated countries are currently (2016) proof-tested at PE piezo pressure.


title: .45 ACP

text: The SAAMI pressure limit for the .45 ACP is set at piezo pressure, while the SAAMI pressure limit for the .45 ACP +P is set at , piezo pressure.


title: .45 ACP

text: The .45 ACP is an effective combat pistol cartridge. It combines accuracy as well as stopping power for use against human targets, has relatively low muzzle blast and flash, and it produces a stout, but manageable recoil in handguns (made worse in compact models).


title: .45 ACP

text: The standard-issue, military .45 ACP cartridge contains a 230-grain bullet that travels at approximately when fired from the government-issue M1911A1 pistol, and approximately fired from the Thompson M1A1 submachine gun. The cartridge comes in various specialty rounds of varying weights and performance levels as well.


title: .45 ACP

text: The cartridge operates at a relatively low maximum chamber pressure rating of (compared to for 9mm Parabellum and .40 S&W, for 10mm Auto, for .357 SIG), which due to a low bolt thrust helps extend service life of weapons in which it is used. Some makers of pistols chambered in .45 ACP do not certify them to use Plus P ammunition.


title: .45 ACP

text: In its non-expanding full metal jacket (FMJ) version, the .45 ACP cartridge has a reputation for effectiveness against human targets because of its heavy mass, having the capacity to penetrate tissue deeply, and damage the central nervous system. Its large 11.5 mm diameter creates a more substantial permanent wound channel versus smaller calibers, which can lower blood pressure rapidly if critical organs of the circulatory system are hit.


title: .45 ACP

text: In its expanding hollow point form, it is also particularly effective against human targets. In tests against ballistic gelatin, a 185-grain hollow point traveling at 1,050 feet per second expanded to about .76 inch. This is a significantly large permanent wound cavity for a handgun projectile. For those who follow the energy dump and/or hydrostatic shock theories of wounding ballistics, this is ideal. While slightly decreasing penetration and likewise the chance of hitting a vital organ, a large diameter wound will cause more blood loss. There is also a reduced likelihood of overpenetration, meaning that it is more likely that the projectile will transfer all of its kinetic energy to the intended target, thus more reliably incapacitating them.


title: .45 ACP

text: Drawbacks for military use include the cartridge's large size, weight, increased material costs in comparison to the smaller, flatter shooting NATO standard 9×19mm Parabellum cartridge, a cartridge which uses less powder, brass, and lead per round. Standard 9mm NATO ammunition has a more limited armor penetration capability—a deficiency shared with .45 ACP, whose large, slow bullet does not penetrate armor to any great extent. The low muzzle velocity also makes the bullet drop over long ranges, making hits more difficult; however, it is important to note that the vast majority of self-defense situations involving handguns typically occur at close ranges.


title: .45 ACP

text: After two years of testing, one of the final FBI comments was that services that adopt (or stay with) .40 S&W or .45 ACP did so at the risk of increased recoil and a possible reduction in accuracy as 9×19mm with premium quality ammunition had nearly exactly the same performance. A factor rated by the recent FBI testing was accuracy and time to recover. The .45 ACP handguns ranked last, largely due to increased recoil.


title: .45 ACP

text: Because of its large diameter and straight-walled design, the .45 ACP geometry is the highest power-per-pressure repeating production round in existence. This is because of the higher powers achievable with .45 ACP +P, .45 Super, and .460 Rowland loads. Because of the inherent low pressure of the standard round, however, compensators and brakes have little effect until +P, Super, and Rowland loads are utilized.


title: .45 ACP

text: As standard pressure .45 ACP rounds fired from handguns and submachine guns are inherently subsonic, it is one of the most powerful pistol calibers available for use in suppressed weapons since subsonic rounds are quieter than supersonic rounds. The latter inevitably produce a highly compressed shock wave, audible as a loud "crack", a small sonic boom, while they travel through the air. Suppressors reduce the audible "report" by slowing and channeling the high speed gas generated by the burning/expanding gunpowder before it exits the muzzle resulting in a muffled "cough". Suppressors cannot act on a supersonic shock wave continuously generated by a bullet exceeding the speed of sound at ambient cold temperatures, as this shock wave is continuously produced throughout the entire flight path over which the bullet is supersonic, which extends long after it exits the barrel.


title: .45 ACP

text: The downside to the use of .45 ACP in suppressed weapons is that increasing the diameter of the passage through a suppressor decreases the suppressor's efficiency; thus, while .45 ACP is among the most powerful suppressed pistol rounds, it is also one of the loudest. Most .45 suppressors must be fired "wet" (with an ablative medium, usually oil or water) to bring sound levels down to "hearing-safe" (under 140 dB, generally).


title: .45 ACP

text: Magazine capacity varies depending on the type of firearm. Standard (not extended) single-stack magazines, pistols based on the 1911 design commonly hold eight rounds or less. Many modern pistols have adapted the cartridge into double-stacked magazine designs to increase ammo capacity, though this increases the pistol's handle girth, but not width. The less-rounded back strap helps to spread the recoil out more and make the pistol more pleasant to shoot. Drum magazines used mostly for submachine guns have a capacity of 50 or 100 rounds.


title: .45 ACP

text: Several US tactical police units still use the .45 pistol round. While high capacity firearms are available in .45 ACP, the greater length and diameter of the .45 ACP means that the grip of the pistol must be longer and wider than the grip of a comparable pistol of a smaller caliber; this increase in grip size can make the pistol difficult to use for shooters with smaller hands.


title: .45 ACP

text: Today, most NATO militaries use sidearms chambered for the 9×19mm Parabellum cartridge, but the effectiveness of the .45 ACP cartridge has ensured its continued popularity with large-caliber sport shooters, especially in the United States. In addition, select military and police units around the world still use firearms firing the .45 ACP. In 1985, the .45 ACP M1911A1 pistol was replaced by the Beretta M9 9mm pistol as the main sidearm of the U.S. military, although select Special Operations units continue to use the M1911A1 or other .45 ACP pistols.


title: .45 ACP

text: Rounds are available from 68 grains to 300 grains (4.4 g to 19 g) with a common load being the standard military loading of a FMJ bullet (for comparison, the most common 9mm load is , half the weight). Specialty rounds are available in weights under and over ; popular rounds among reloaders and target shooters include 185-grain to 230-grain (12 g to 15 g) bullets. Target shooters competing in Bullseye Pistol (aka Precision Pistol) find that .45 ACP ammunition using light bullets (12 to 13 g) and low velocities paradoxically generates less recoil than 9mm ammunition of equivalent accuracy, despite its larger caliber, and allows better scores in sustained fire. Hollow-point rounds intended for maximum effectiveness against live targets are designed to expand upon impact with soft tissue, increasing the size of the permanent cavity left by the bullet as it passes through the target.


title: .45 ACP

text: Tracer ammunition for the .45 ACP was manufactured by Frankford Arsenal and by Remington Arms. This ammunition was available to the United States Border Patrol as early as 1940 and was used through World War II for emergency signalling by downed United States Navy and Marine Corps air crew. Tracer ammunition was identified by painting the bullet tip red.


title: .45 ACP

text: Most ammunition manufacturers also market what are termed "+P" (pronounced "plus P", designating overpressure ammunition) loadings in pistol ammunition, including the .45 ACP. This means the cartridge is loaded to a higher maximum pressure level than the original SAAMI cartridge standard, generating higher velocity and more muzzle energy. In the case of the .45 ACP, the new standard cartridge pressure is and the SAAMI .45 ACP +P standard is . This is a common practice for updating older cartridges to match the better quality of materials and workmanship in modern firearms.


title: .45 ACP

text: The terminology is generally given as "45 ACP +P" and sometimes, but not always, appears on the headstamp. These cartridges have the same external dimensions as the standard-pressure cartridges and will chamber and fire in all firearms designed for the standard-pressure loadings. The inner dimensions of the +P cartridge are different from the standard-pressure cartridge dimensions and thus allows for higher pressures to be safely achieved in the +P cartridge. If +P loadings are used in firearms not specifically designed for them, they may cause damage to the weapon and injuries to the operator.


title: .45 ACP

text: Popular derivative versions of the .45 ACP are .45 Super and .460 Rowland. The Super is dimensionally identical to the .45 ACP; however, the cartridge carries a developer established pressure of and requires minor modification of firearms for use. The Rowland operates at a developer established SAAMI and may only be used within a select group of firearms significantly modified for this purpose; the Rowland case is longer specifically to prevent it from being chambered in standard .45 ACP firearms. Brass cases for each of these cartridges carry the applicable name within the headstamp. The Super provides approximately 20% greater velocity than the .45 ACP +P; the Rowland approximately 40% greater velocity than the .45 ACP +P.


title: Integral

text: In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.


title: Integral

text: The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.


title: Integral

text: Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.


title: Integral

text: Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.


title: Integral

text: The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ("ca." 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.


title: Integral

text: A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.


title: Integral

text: In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( AD) derived a formula for the sum of fourth powers. 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.


title: Integral

text: The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of Indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of up to degree in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of to a general power, including negative powers and fractional powers.


title: Integral

text: The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.


title: Integral

text: While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.


title: Integral

text: The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol, ∫, from the letter "ſ" (long s), standing for "summa" (written as "ſumma"; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in "Mémoires" of the French Academy around 1819–20, reprinted in his book of 1822.


title: Integral

text: Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.


title: Integral

text: The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur".


title: Integral

text: In general, the integral of a real-valued function with respect to a real variable on an interval is written as


title: Integral

text: The integral sign represents integration. The symbol , called the differential of the variable , indicates that the variable of integration is . The function is called the integrand, the points and are called the limits (or bounds) of integration, and the integral is said to be over the interval , called the interval of integration.


title: Integral

text: A function is said to be if its integral over its domain is finite. If limits are specified, the integral is called a definite integral.


title: Integral

text: the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).


title: Integral

text: In advanced settings, it is not uncommon to leave out when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write formula_3 to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.


title: Integral

text: Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation.


title: Integral

text: For example, to find the area of the region bounded by the graph of the function between and , one can cross the interval in five steps (), then fill a rectangle using the right end height of each piece (thus ) and sum their areas to get an approximation of


title: Integral

text: which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increase to infinity, it will reach a limit which is the exact value of the area sought (in this case, ). One writes


title: Integral

text: which means is the result of a weighted sum of function values, , multiplied by the infinitesimal step widths, denoted by , on the interval .


title: Integral

text: There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.


title: Integral

text: The Riemann integral is defined in terms of Riemann sums of functions with respect to "tagged partitions" of an interval. A tagged partition of a closed interval on the real line is a finite sequence


title: Integral

text: This partitions the interval into sub-intervals indexed by , each of which is "tagged" with a distinguished point . A "Riemann sum" of a function with respect to such a tagged partition is defined as


title: Integral

text: thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the width of sub-interval, . The "mesh" of such a tagged partition is the width of the largest sub-interval formed by the partition, . The "Riemann integral" of a function over the interval is equal to if:


title: Integral

text: When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.


title: Integral

text: It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated.


title: Integral

text: Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:


title: Integral

text: As Folland puts it, "To compute the Riemann integral of , one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of ". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure of an interval is its width, , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.


title: Integral

text: Using the "partitioning the range of " philosophy, the integral of a non-negative function should be the sum over of the areas between a thin horizontal strip between and . This area is just . Let . The Lebesgue integral of is then defined by


title: Integral

text: where the integral on the right is an ordinary improper Riemann integral ( is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines the Lebesgue integral.


title: Integral

text: A general measurable function is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of and the -axis is finite:


title: Integral

text: In that case, the integral is, as in the Riemannian case, the difference between the area above the -axis and the area below the -axis:


title: Integral

text: Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:


title: Integral

text: The collection of Riemann-integrable functions on a closed interval forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration


title: Integral

text: is a linear functional on this vector space. Thus, the collection of integrable functions is closed under taking linear combinations, and the integral of a linear combination is the linear combination of the integrals:


title: Integral

text: Similarly, the set of real-valued Lebesgue-integrable functions on a given measure space with measure is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral


title: Integral

text: More generally, consider the vector space of all measurable functions on a measure space , taking values in a locally compact complete topological vector space over a locally compact topological field . Then one may define an abstract integration map assigning to each function an element of or the symbol ,


title: Integral

text: that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of (i.e. "finite"). The most important special cases arise when is , , or a finite extension of the field of p-adic numbers, and is a finite-dimensional vector space over , and when and is a complex Hilbert space.


title: Integral

text: Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set , generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See for an axiomatic characterization of the integral.


title: Integral

text: A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval and can be generalized to other notions of integral (Lebesgue and Daniell).


title: Integral

text: over an interval is defined if . This means that the upper and lower sums of the function are evaluated on a partition whose values are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating within intervals where an interval with a higher index lies to the right of one with a lower index. The values and , the end-points of the interval, are called the limits of integration of . Integrals can also be defined if :""


title: Integral

text: The first convention is necessary in consideration of taking integrals over subintervals of ; the second says that an integral taken over a degenerate interval, or a point, should be zero. One reason for the first convention is that the integrability of on an interval implies that is integrable on any subinterval , but in particular integrals have the property that if is any element of , then:""


title: Integral

text: The "fundamental theorem of calculus" is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the "second fundamental theorem of calculus", allows one to compute integrals by using an antiderivative of the function to be integrated.


title: Integral

text: Let be a continuous real-valued function defined on a closed interval . Let be the function defined, for all in , by


title: Integral

text: Let be a real-valued function defined on a closed interval [] that admits an antiderivative on . That is, and are functions such that for all in ,


title: Integral

text: A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.


title: Integral

text: If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:


title: Integral

text: If the integrand is only defined or finite on a half-open interval, for instance , then again a limit may provide a finite result:


title: Integral

text: That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or , or . In more complicated cases, limits are required at both endpoints, or at interior points.


title: Integral

text: Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the "x"-axis, the "double integral" of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains its domain. For example, a function in two dimensions depends on two real variables, "x" and "y", and the integral of a function "f" over the rectangle "R" given as the Cartesian product of two intervals formula_42 can be written


title: Integral

text: where the differential indicates that integration is taken with respect to area. This double integral can be defined using Riemann sums, and represents the (signed) volume under the graph of over the domain "R". Under suitable conditions (e.g., if "f" is continuous), Fubini's theorem states that this integral can be expressed as an equivalent iterated integral


title: Integral

text: This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over "R" uses a double integral sign:


title: Integral

text: Integration over more general domains is possible. The integral of a function "f", with respect to volume, over an "n-"dimensional region "D" of formula_46 is denoted by symbols such as:


title: Integral

text: The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields.


title: Integral

text: A "line integral" (sometimes called a "path integral") is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a "contour integral".


title: Integral

text: The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force, , multiplied by displacement, , may be expressed (in terms of vector quantities) as:


title: Integral

text: For an object moving along a path in a vector field such as an electric field or gravitational field, the total work done by the field on the object is obtained by summing up the differential work done in moving from to . This gives the line integral


title: Integral

text: A "surface integral" generalizes double integrals to integration over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a vector field. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.


title: Integral

text: For an example of applications of surface integrals, consider a vector field on a surface ; that is, for each point in , is a vector. Imagine that a fluid flows through , such that determines the velocity of the fluid at . The flux is defined as the quantity of fluid flowing through in unit amount of time. To find the flux, one need to take the dot product of with the unit surface normal to at each point, which will give a scalar field, which is integrated over the surface:


title: Integral

text: The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the classical theory of electromagnetism.


title: Integral

text: In complex analysis, the integrand is a complex-valued function of a complex variable instead of a real function of a real variable . When a complex function is integrated along a curve formula_51 in the complex plane, the integral is denoted as follows


title: Integral

text: A differential form is a mathematical concept in the fields of multivariable calculus, differential topology, and tensors. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:


title: Integral

text: where "E", "F", "G" are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials "dx", "dy", "dz" measure infinitesimal oriented lengths parallel to the three coordinate axes.


title: Integral

text: Here the basic two-forms formula_55 measure oriented areas parallel to the coordinate two-planes. The symbol formula_56 denotes the wedge product, which is similar to the cross product in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of formula_57.


title: Integral

text: Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.


title: Integral

text: The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time-scale calculus.


title: Integral

text: An integration that is performed not over a variable (or, in physics, over a space or time dimension), but over a space of functions, is referred to as a functional integral.


title: Integral

text: Integrals are used extensively in many areas. For example, in probability theory, integrals are used to determine the probability of some random variable falling within a certain range. Moreover, the integral under an entire probability density function must equal 1, which provides a test of whether a function with no negative values could be a density function or not.


title: Integral

text: Integrals can be used for computing the area of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral. The volume of a three-dimensional object such as a disc or washer can be computed by disc integration using the equation for the volume of a cylinder, formula_58, where formula_59 is the radius. In the case of a simple disc created by rotating a curve about the -axis, the radius is given by , and its height is the differential . Using an integral with bounds and , the volume of the disc is equal to:formula_60Integrals are also used in physics, in areas like kinematics to find quantities like displacement, time, and velocity. For example, in rectilinear motion, the displacement of an object over the time interval formula_61 is given by:


title: Integral

text: where formula_63 is the velocity expressed as a function of time. The work done by a force formula_64 (given as a function of position) from an initial position formula_65 to a final position formula_66 is:


title: Integral

text: Integrals are also used in thermodynamics, where thermodynamic integration is used to calculate the difference in free energy between two given states.


title: Integral

text: The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. Let be the function of to be integrated over a given interval . Then, find an antiderivative of ; that is, a function such that on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus,


title: Integral

text: Sometimes it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include integration by substitution, integration by parts, integration by trigonometric substitution, and integration by partial fractions.


title: Integral

text: Alternative methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.


title: Integral

text: Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.


title: Integral

text: Specific results which have been worked out by various techniques are collected in the list of integrals.


title: Integral

text: Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like Macsyma and Maple.


title: Integral

text: A major mathematical difficulty in symbolic integration is that in many cases, a relatively simple function does not have integrals that can be expressed in closed form involving only elementary functions, include rational and exponential functions, logarithm, trigonometric functions and inverse trigonometric functions, and the operations of multiplication and composition. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and to compute it if it is. However, functions with closed expressions of antiderivatives are the exception, and consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in Mathematica, Maple and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions.


title: Integral

text: Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject.


title: Integral

text: More recently a new approach has emerged, using "D"-finite functions, which are the solutions of linear differential equations with polynomial coefficients. Most of the elementary and special functions are "D"-finite, and the integral of a "D"-finite function is also a "D"-finite function. This provides an algorithm to express the antiderivative of a "D"-finite function as the solution of a differential equation. This theory also allows one to compute the definite integral of a "D"-function as the sum of a series given by the first coefficients, and provides an algorithm to compute any coefficient.


title: Integral

text: Definite integrals may be approximated using several methods of numerical integration. The rectangle method relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the trapezoidal rule, replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation. The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: Simpson's rule approximates the integrand by a piecewise quadratic function.


title: Integral

text: Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the Newton–Cotes formulas. The degree Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree "" polynomial. This polynomial is chosen to interpolate the values of the function on the interval. Higher degree Newton–Cotes approximations can be more accurate, but they require more function evaluations, and they can suffer from numerical inaccuracy due to Runge's phenomenon. One solution to this problem is Clenshaw–Curtis quadrature, in which the integrand is approximated by expanding it in terms of Chebyshev polynomials.


title: Integral

text: Romberg's method halves the step widths incrementally, giving trapezoid approximations denoted by , , and so on, where is half of . For each new step size, only half the new function values need to be computed; the others carry over from the previous size. It then interpolate a polynomial through the approximations, and extrapolate to . Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. An -point Gaussian method is exact for polynomials of degree up to .