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Id Software

id Software LLC is an American video game developer based in Richardson, Texas. It was founded on February 1, 1991, by four members of the computer company Softdisk: programmers John Carmack and John Romero, game designer Tom Hall, and artist Adrian Carmack.

id Software made important technological developments in video game technologies for the PC (running MS-DOS and Windows), including work done for the Wolfenstein, Doom, and Quake franchises at the time. id's work was particularly important in 3D computer graphics technology and in game engines that are used throughout the video game industry. The company was involved in the creation of the first-person shooter (FPS) genre: Wolfenstein 3D is often considered to be the first true FPS; Doom is a game that popularized the genre and PC gaming in general; and Quake was id's first true 3D FPS.

On June 24, 2009, ZeniMax Media acquired the company. In 2015, they opened a second studio in Frankfurt, Germany.

Formation
The founders of id Software – John Carmack, John Romero, and Tom Hall – met in the offices of Softdisk developing multiple games for Softdisk's monthly publishing, including Dangerous Dave. Along with another Softdisk employee, Lane Roathe, they had formed a small group they called Ideas from the Deep (IFD), a name that Romero and Roathe had come up with. In September 1990, Carmack developed an efficient way to rapidly side-scroll graphics on the PC. Upon making this breakthrough, Carmack and Hall stayed up late into the night making a replica of the first level of the popular 1988 NES game Super Mario Bros. 3, inserting stock graphics of Romero's Dangerous Dave character in lieu of Mario. When Romero saw the demo, entitled Dangerous Dave in Copyright Infringement, he realized that Carmack's breakthrough could have potential. The IFD team moonlighted over a week and over two weekends to create a larger demo of their PC version of Super Mario Bros. 3. They sent their work to Nintendo. According to Romero, Nintendo had told them that the demo was impressive, but "they didn't want their intellectual property on anything but their own hardware, so they told us Good Job and You Can't Do This". While the pair had not readily shared the demo though acknowledged its existence in the years since, a working copy of the demo was discovered in July 2021 and preserved at the Museum of Play.

Around the same time in 1990, Scott Miller of Apogee Software learned of the group and their exceptional talent, having played one of Romero's Softdisk games, Dangerous Dave, and contacted Romero under the guise of multiple fan letters that Romero came to realize all originated from the same address. When he confronted Miller, Miller explained that the deception was necessary since Softdisk screened letters it received. Although disappointed by not actually having received mail from multiple fans, Romero and other Softdisk developers began proposing ideas to Miller. One of these was Commander Keen, a side-scrolling game that incorporated the previous work they had done on the Super Mario Bros. 3 demonstration. The first Commander Keen game, Commander Keen in Invasion of the Vorticons, was released through Apogee in December 1990, which became a very successful shareware game. After their first royalty check, Romero, Carmack, and Adrian Carmack (no relation) decided to start their own company. After hiring Hall, the group finished the Commander Keen series, then hired Jay Wilbur and Kevin Cloud and began working on Wolfenstein 3D. id Software was officially founded by Romero, John and Adrian Carmack and Hall on February 1, 1991. The name "id" came out of their previous IFD; Roathe had left the group, and they opted to drop the "F" to leave "id". They initially used "id" as an initialism for "In Demand", but by the time of the fourth Commander Keen game, they opted to let "id" stand out "as a cool word", according to Romero.

The shareware distribution method was initially employed by id Software through Apogee Software to sell their products, such as the Commander Keen, Wolfenstein and Doom games. They would release the first part of their trilogy as shareware, then sell the other two installments by mail order. Only later (about the time of the release of Doom II) did id Software release their games via more traditional shrink-wrapped boxes in stores (through other game publishers).

After Wolfenstein 3D's great success, id began working on Doom. After Hall left the company, Sandy Petersen and Dave Taylor were hired before the release of Doom in December 1993.

The end of the classic lineup
Quake was released on June 22, 1996 and was considered a difficult game to develop due to creative differences. Animosity grew within the company and it caused a conflict between Carmack and Romero, which led the latter to leave id after the game's release. Soon after, other staff left the company as well such as Michael Abrash, Shawn Green, Jay Wilbur, Petersen and Mike Wilson. Petersen claimed in July 2021 that the lack of a team leader was the cause of it all. In fact, he volunteered to take lead as he had five years of experience as project manager in MicroProse but he was turned down by Carmack.

ZeniMax Media and Microsoft
On June 24, 2009, it was announced that id Software had been acquired by ZeniMax Media (owner of Bethesda Softworks). The deal would eventually affect publishing deals id Software had before the acquisition, namely Rage, which was being published through Electronic Arts. ZeniMax received in July a $105 million investment from StrongMail Systems for the id acquisition, it's unknown if that was the exact price of the deal. id Software moved from the "cube-shaped" Mesquite office to a location in Richardson, Texas during the spring of 2011.

On June 26, 2013, id Software president Todd Hollenshead quit after 17 years of service.

On November 22, 2013, it was announced id Software co-founder and Technical Director John Carmack had fully resigned from the company to work full-time at Oculus VR which he joined as CTO in August 2013. He was the last of the original founders to leave the company.

Tim Willits left the company in 2019. ZeniMax Media was acquired by Microsoft for US$7.5 billion in March 2021 and became part of Xbox Game Studios.

Company name
The company writes its name with a lowercase id, which is pronounced as in "did" or "kid", and, according to the book Masters of Doom, the group identified itself as "Ideas from the Deep" in the early days of Softdisk but that, in the end, the name 'id' came from the phrase "in demand". Disliking "in demand" as "lame", someone suggested a connection with Sigmund Freud's psychological concept of id, which the others accepted. Evidence of the reference can be found as early as Wolfenstein 3D with the statement "that's id, as in the id, ego, and superego in the psyche" appearing in the game's documentation. Prior to an update to the website, id's History page made a direct reference to Freud.

Key employees

 * Kevin Cloud – Artist (1992–2006), Executive producer (2007–present)
 * Donna Jackson – Office manager / "id mom" (1994–present)
 * Marty Stratton – Director of Business Development (1997–2006), Executive Producer (2006–present) Studio Director (2019–present)
 * Hugo Martin – Creative Director (2013–present)

Former key employees
Arranged in chronological order:
 * Tom Hall – Co-founder, game designer, level designer, writer, creative director (1991–1993). After a dispute with John Carmack over the designs of Doom, Hall was forced to resign from id Software in August 1993. He joined 3D Realms soon afterwards.
 * Bobby Prince – Music composer (1991–1994). A freelance musician who went on to pursue other projects after Doom II.
 * Dave Taylor – Programmer (1993–1996). Taylor left id Software and co-founded Crack dot Com.
 * John Romero – Co-founder, game designer, programmer (1991–1996). Romero resigned on August 6, 1996. He established Ion Storm along with Hall on November 15, 1996.
 * Michael Abrash – Programmer (1995–1996). Returned to Microsoft after the release of Quake, but eventually worked with Carmack again at Reality Labs.
 * Shawn Green – Software support (1991–1996). Left id Software to join Romero at Ion Storm.
 * Jay Wilbur – Business manager (1991–1997). Left id Software after Romero's departure and joined Epic Games in 1997.
 * Sandy Petersen – Level designer (1993–1997). Left id Software for Ensemble Studios in 1997.
 * Mike Wilson – PR and marketing (1994–1997). Left id Software to become CEO of Ion Storm with Romero. Left a year later to found Gathering of Developers and later Devolver Digital.
 * American McGee – Level designer (1993–1998). McGee was fired after the release of Quake II. He joined Electronic Arts and created American McGee's Alice.
 * Adrian Carmack – Co-founder, artist (1991–2005). Carmack was forced out of id Software after the release of Doom 3 because he would not sell his stock at a low price to the other owners. Adrian sued id Software and the lawsuit was settled during the Zenimax acquisition in 2009.
 * Todd Hollenshead – President (1996–2013) Left id Software on good terms to work at Nerve Software.
 * John Carmack – Co-founder, technical director (1991–2013). He joined Oculus VR on August 7, 2013, as a side project, but unable to handle two companies at the same time, Carmack resigned from id Software on November 22, 2013, to pursue Oculus full-time, making him the last founding member to leave the company.
 * Tim Willits – Level designer (1995–2001), creative director (2002–2011), studio director (2012–2019) He is now the chief creative officer at Saber Interactive.
 * Robert Duffy – Chief Technology Officer (1998–2024). Robert left id Software in January 2024.

Technology
Starting with their first shareware game series, Commander Keen, id Software has licensed the core source code for the game, or what is more commonly known as the engine. Brainstormed by John Romero, id Software held a weekend session titled "The id Summer Seminar" in the summer of 1991 with prospective buyers including Scott Miller, George Broussard, Ken Rogoway, Jim Norwood and Todd Replogle. One of the nights, id Software put together an impromptu game known as "Wac-Man" to demonstrate not only the technical prowess of the Keen engine, but also how it worked internally.

id Software has developed their own game engine for each of their titles when moving to the next technological milestone, including Commander Keen, Wolfenstein 3D, ShadowCaster, Doom, Quake, Quake II, and Quake III, as well as the technology used in making Doom 3. After being used first for id Software's in-house game, the engines are licensed out to other developers. According to Eurogamer.net, "id Software has been synonymous with PC game engines since the concept of a detached game engine was first popularized". During the mid to late 1990s, "the launch of each successive round of technology it's been expected to occupy a headlining position", with the Quake III engine being most widely adopted of their engines. However id Tech 4 had far fewer licensees than the Unreal Engine from Epic Games, due to the long development time that went into Doom 3 which id Software had to release before licensing out that engine to others.

Despite his enthusiasm for open source code, Carmack revealed in 2011 that he had no interest in licensing the technology to the mass market. Beginning with Wolfenstein 3D, he felt bothered when third-party companies started "pestering" him to license the id tech engine, adding that he wanted to focus on new technology instead of providing support to existing ones. He felt very strongly that this was not why he signed up to be a game programmer for; to be "holding the hands" of other game developers. Carmack commended Epic Games for pursuing the licensing to the market beginning with Unreal Engine 3. Even though the said company has gained more success with its game engine than id Software over the years, Carmack had no regrets by his decision and continued to focus on open source until his departure from the company in 2013.

In conjunction with his self-professed affinity for sharing source code, John Carmack has open-sourced most of the major id Software engines under the GNU General Public License. Historically, the source code for each engine has been released once the code base is 5 years old. Consequently, many home grown projects have sprung up porting the code to different platforms, cleaning up the source code, or providing major modifications to the core engine. Wolfenstein 3D, Doom and Quake engine ports are ubiquitous to nearly all platforms capable of running games, such as hand-held PCs, iPods, the PSP, the Nintendo DS and more. Impressive core modifications include GZDoom, which adds to the Doom engine modern hardware accelerared renderers and a scripting system called ZScript, and was also utilized in the creation of ECWolf for Wolfenstein 3D  and Raze for the Build engine. Meanwhile DarkPlaces adds stencil shadow volumes into the original Quake engine along with a more efficient network protocol. Other projects include Yamagi Quake II, ioquake3, and dhewm3, which maintain the goal of cleaning up the source code, adding features and fixing bugs. Even earlier id Software code, namely for Hovertank 3D and Catacomb 3D, was released in June 2014 by Flat Rock Software.

The GPL release of the Quake III engine's source code was moved from the end of 2004 to August 2005 as the engine was still being licensed to commercial customers who would otherwise be concerned over the sudden loss in value of their recent investment.

On August 4, 2011, John Carmack revealed during his QuakeCon 2011 keynote that they will be releasing the source code of the Doom 3 engine (id Tech 4) during the year.

id Software publicly stated they would not support the Wii console (possibly due to technical limitations), although they have since indicated that they may release titles on that platform (although it would be limited to their games released during the 1990s). They continued this policy with the Wii U but for Nintendo Switch, they collaborated with Panic Button starting with 2016's Doom and Wolfenstein II: The New Colossus.

Since id Software revealed their engine id Tech 5, they call their engines "id Tech", followed by a version number. Older engines have retroactively been renamed to fit this scheme, with the Doom engine as id Tech 1.

IMF Music File Format
IMF ("id music file" or "id's music format") is an audio file format created by id Software for the AdLib sound card for use in their video games. The format is similar to MIDI, in that it defines musical notes, and does not support sampled digital audio for sound effects. IMF files store the actual bytes sent to the AdLib's OPL2 chip, which uses FM synthesis to produce audio output. The format is based on the AdLib command syntax, with a few modifications. Due to the limited features and relatively low sound quality, modern games no longer use IMF music.

A large number of songs in id Software's early games (such as Commander Keen and Wolfenstein 3D) were composed by Bobby Prince in IMF format. Other game developers like Apogee Software also used this format in their games (such as Cosmo's Cosmic Adventure, Duke Nukem II, and Monster Bash).

Linux gaming
id Software was an early pioneer in the Linux gaming market, and id Software's Linux games have been some of the most popular of the platform. Many id Software games won the Readers' and Editors' Choice awards of Linux Journal. Some id Software titles ported to Linux are Doom (the first id Software game to be ported), Quake, Quake II, Quake III Arena, Return to Castle Wolfenstein, Wolfenstein: Enemy Territory, Doom 3, Quake 4, and Enemy Territory: Quake Wars. Since id Software and some of its licensees released the source code for some of their previous games, several games which were not ported (such as Catacomb 3D, Catacomb Abyss, Wolfenstein 3D, Spear of Destiny, Blake Stone: Aliens of Gold, Blake Stone: Planet Strike, Super 3D Noah's Ark, Rise of the Triad, Doom 64, Strife, Heretic, Hexen, Hexen II, Star Trek: Voyager – Elite Force Holomatch, Star Wars Jedi Knight II: Jedi Outcast, Star Wars Jedi Knight: Jedi Academy) can run on Linux and other operating systems natively through the use of source ports. Quake Live also launched with Linux support, although this, alongside OS X support, was later removed when changed to a standalone title.

The tradition of porting to Linux was first started by Dave D. Taylor, with Zoid Kirsch doing some later porting. Since Quake III Arena, Linux porting had been handled by Timothee Besset. The majority of all id Tech 4 games, including those made by other developers, have a Linux client available, the only current exceptions being Wolfenstein and Brink. Similarly, almost all of the games utilizing the Quake II engine have Linux ports, the only exceptions being those created by Ion Storm (Daikatana later received a community port). Despite fears by the Linux gaming community that id Tech 5 would not be ported to that platform, Timothee Besset in his blog stated "I'll be damned if we don't find the time to get Linux builds done". Besset explained that id Software's primary justification for releasing Linux builds was better code quality, along with a technical interest in the platform. However, on January 26, 2012, Besset announced that he had left id.

John Carmack has expressed his stance with regard to Linux builds in the past. In December 2000 Todd Hollenshead expressed support for Linux: "All said, we will continue to be a leading supporter of the Linux platform because we believe it is a technically sound OS and is the OS of choice for many server ops." However, on April 25, 2012, Carmack revealed that "there are no plans for a native Linux client" of id's most recent game, Rage. In February 2013, Carmack argued for improving emulation as the "proper technical direction for gaming on Linux", though this was also due to ZeniMax's refusal to support "unofficial binaries", given all prior ports (except for Quake III Arena, via Loki Software, and earlier versions of Quake Live) having only ever been unofficial. Carmack didn't mention official games Quake: The Offering and Quake II: Colossus ported by id Software to Linux and published by Macmillan Computer Publishing USA.

Despite no longer releasing native binaries, id was an early adopter of Stadia, a cloud gaming service powered by Debian Linux servers, and the cross-platform Vulkan API. A Linux version of Doom from 2016 was tested internally, while it and its sequel Doom Eternal can be run using Wine and Proton.

Commander Keen
Commander Keen in Invasion of the Vorticons, a platform game in the style of those for the Nintendo Entertainment System, was one of the first MS-DOS games with smooth horizontal-scrolling. Published by Apogee Software, the title and follow-ups brought id Software success as a shareware developer. It is the series of id Software that designer Tom Hall is most affiliated with. The first Commander Keen trilogy was released on December 14, 1990.

Wolfenstein
The company's breakout product was released on May 5, 1992: Wolfenstein 3D, a first-person shooter (FPS) with smooth 3D graphics that were unprecedented in computer games, and with violent gameplay that many gamers found engaging. After essentially founding an entire genre with this game, id Software created Doom, Doom II: Hell on Earth, Quake, Quake II, Quake III Arena, Quake 4, and Doom 3. Each of these first-person shooters featured progressively higher levels of graphical technology. Wolfenstein 3D spawned a prequel and a sequel: the prequel called Spear of Destiny, and the second, Return to Castle Wolfenstein, using the id Tech 3 engine. A third Wolfenstein sequel, simply titled Wolfenstein, was released by Raven Software, using the id Tech 4 engine. Another sequel, named Wolfenstein: The New Order; was developed by MachineGames using the id Tech 5 engine and released in 2014, with it getting a prequel by the name of Wolfenstein: The Old Blood a year later; followed by a direct sequel titled Wolfenstein II: The New Colossus in 2017.

Doom
Eighteen months after their release of Wolfenstein 3D, on December 10, 1993, id Software released Doom which would again set new standards for graphic quality and graphic violence in computer gaming. Doom featured a sci-fi/horror setting with graphic quality that had never been seen on personal computers or even video game consoles. Doom became a cultural phenomenon and its violent theme would eventually launch a new wave of criticism decrying the dangers of violence in video games. Doom was ported to numerous platforms, inspired many knock-offs, and was eventually followed by the technically similar Doom II: Hell on Earth. id Software made its mark in video game history with the shareware release of Doom, and eventually revisited the theme of this game in 2004 with their release of Doom 3. John Carmack said in an interview at QuakeCon 2007 that there would be a Doom 4. It began development on May 7, 2008. Doom 2016, the fourth installation of the Doom series, was released on Microsoft Windows, PlayStation 4, and Xbox One on May 13, 2016, and was later released on Nintendo Switch on November 10, 2017. In June 2018, the sequel to the 2016 Doom, Doom Eternal was officially announced at E3 2018 with a teaser trailer, followed by a gameplay reveal at QuakeCon in August 2018.

Quake
On June 22, 1996, the release of Quake marked the third milestone in id Software history. Quake combined a cutting edge fully 3D engine, the Quake engine, with a distinctive art style to create critically acclaimed graphics for its time. Audio was not neglected either, having recruited Nine Inch Nails frontman Trent Reznor to facilitate unique sound effects and ambient music for the game. (A small homage was paid to Nine Inch Nails in the form of the band's logo appearing on the ammunition boxes for the nailgun weapon.) It also included the work of Michael Abrash. Furthermore, Quake's main innovation, the capability to play a deathmatch (competitive gameplay between living opponents instead of against computer-controlled characters) over the Internet (especially through the add-on QuakeWorld), seared the title into the minds of gamers as another smash hit.

In 2008, id Software was honored at the 59th Annual Technology & Engineering Emmy Awards for the pioneering work Quake represented in user modifiable games. id Software is the only game development company ever honored twice by the National Academy of Television Arts & Sciences, having been given an Emmy Award in 2007 for creation of the 3D technology that underlies modern shooter video games.

The Quake series continued with Quake II in 1997. Activision purchased a 49% stake in id Software, making it a second party which took publishing duties until 2009. However, the game is not a storyline sequel, and instead focuses on an assault on an alien planet, Stroggos, in retaliation for Strogg attacks on Earth. Most of the subsequent entries in the Quake franchise follow this storyline. Quake III Arena (1999), the next title in the series, has minimal plot, but centers around the "Arena Eternal", a gladiatorial setting created by an alien race known as the Vadrigar and populated by combatants plucked from various points in time and space. Among these combatants are some characters either drawn from or based on those in Doom ("Doomguy"), Quake (Ranger, Wrack), and Quake II (Bitterman, Tank Jr., Grunt, Stripe). Quake IV (2005) picks up where Quake II left off – finishing the war between the humans and Strogg. The spin-off Enemy Territory: Quake Wars acts as a prequel to Quake II, when the Strogg first invade Earth. Quake IV and Enemy Territory: Quake Wars were made by outside developers and not id.

There have also been other spin-offs such as Quake Mobile in 2005 and Quake Live, an internet browser based modification of Quake III. A game called Quake Arena DS was planned and canceled for the Nintendo DS. John Carmack stated, at QuakeCon 2007, that the id Tech 5 engine would be used for a new Quake game.

Rage
Todd Hollenshead announced in May 2007 that id Software had begun working on an all new series that would be using a new engine. Hollenshead also mentioned that the title would be completely developed in-house, marking the first game since 2004's Doom 3 to be done so. At 2007's WWDC, John Carmack showed the new engine called id Tech 5. Later that year, at QuakeCon 2007, the title of the new game was revealed as Rage.

On July 14, 2008, id Software announced at the 2008 E3 event that they would be publishing Rage through Electronic Arts, and not id's longtime publisher Activision. However, since then ZeniMax has also announced that they are publishing Rage through Bethesda Softworks.

On August 12, 2010, during Quakecon 2010, id Software announced Rage US ship date of September 13, 2011, and a European ship date of September 15, 2011. During the keynote, id Software also demonstrated a Rage spin-off title running on the iPhone. This technology demo later became Rage HD. The game was ultimately released in October 2011.

On May 14, 2018, Bethesda Softworks announced Rage 2, a co-development between id Software and Avalanche Studios.

Other games
During its early days, id Software produced much more varied games; these include the early 3D first-person shooter experiments that led to Wolfenstein 3D and Doom – Hovertank 3D and Catacomb 3D. There was also the Rescue Rover series, which had two games – Rescue Rover and Rescue Rover 2. Also there was John Romero's Dangerous Dave series, which included such notables as the tech demo (In Copyright Infringement) which led to the Commander Keen engine, and the decently popular Dangerous Dave in the Haunted Mansion. In the Haunted Mansion was powered by the same engine as the earlier id Software game Shadow Knights, which was one of the several games written by id Software to fulfill their contractual obligation to produce games for Softdisk, where the id Software founders had been employed. id Software has also overseen several games using its technology that were not made in one of their IPs such as ShadowCaster, (early-id Tech 1), Heretic, Hexen: Beyond Heretic (id Tech 1), Hexen II (Quake engine), and Orcs and Elves (Doom RPG engine).

Other media
id Software has also published novels based on the Doom series Doom novels. After a brief hiatus from publishing, id resumed and re-launched the novel series in 2008 with Matthew J. Costello's (a story consultant for Doom 3 and now Rage) new Doom 3 novels: Worlds on Fire and Maelstrom.

id Software became involved in film development when they oversaw the film adaption of their Doom franchise in 2005. In August 2007, Todd Hollenshead stated at QuakeCon 2007 that a Return to Castle Wolfenstein movie is in development which re-teams the Silent Hill writer/producer team, Roger Avary as writer and director and Samuel Hadida as producer. A new Doom film, titled Doom: Annihilation, was released in 2019, although id itself stressed its lack of involvement.

Controversy
id Software was the target of controversy over two of their most popular games, Doom and the earlier Wolfenstein 3D. More recently in 2022, id Software found themselves mired in a controversy concerning libel against Doom Eternal's composer.

Doom
Doom was notorious for its high levels of gore and occultism along with satanic imagery, which generated controversy from a broad range of groups. Yahoo! Games listed it as one of the top ten most controversial games of all time.

The game again sparked controversy throughout a period of school shootings in the United States when it was found that Eric Harris and Dylan Klebold, who committed the Columbine High School massacre in 1999, were avid players of the game. While planning for the massacre, Harris said that the killing would be "like playing Doom", and "it'll be like the LA riots, the Oklahoma bombing, World War II, Vietnam, Duke Nukem and Doom all mixed together", and that his shotgun was "straight out of the game". A rumor spread afterwards that Harris had designed a Doom level that looked like the high school, populated with representations of Harris's classmates and teachers, and that Harris practiced for his role in the shootings by playing the level over and over. Although Harris did design Doom levels, none of them were based on Columbine High School.

While Doom and other violent video games have been blamed for nationally covered school shootings, 2008 research featured by Greater Good Science Center shows that the two are not closely related. Harvard Medical School researchers Cheryl Olson and Lawrence Kutner found that violent video games did not correlate to school shootings. The United States Secret Service and United States Department of Education analyzed 37 incidents of school violence and sought to develop a profile of school shooters; they discovered that the most common traits among shooters were that they were male and had histories of depression and attempted suicide. While many of the killers—like the vast majority of young teenage boys—did play video games, this study did not find a relationship between gameplay and school shootings. In fact, only one-eighth of the shooters showed any special interest in violent video games, far less than the number of shooters who seemed attracted to books and movies with violent content.

Wolfenstein 3D
As for Wolfenstein 3D, due to its use of Nazi symbols such as the swastika and the anthem of the Nazi Party, Horst-Wessel-Lied, as theme music, the PC version of the game was withdrawn from circulation in Germany in 1994, following a verdict by the Amtsgericht München on January 25, 1994. Despite the fact that Nazis are portrayed as the enemy in Wolfenstein, the use of those symbols is a federal offense in Germany unless certain circumstances apply. Similarly, the Atari Jaguar version was confiscated following a verdict by the Amtsgericht Berlin Tiergarten on December 7, 1994. The Unterhaltungssoftware Selbstkontrolle lifted the outright ban in 2018 in favor of analysing depictions on a case-by-case basis, and the international version of the game was removed from the list of banned titles in 2019.

Due to concerns from Nintendo of America, the Super NES version was modified to not include any swastikas or Nazi references; furthermore, blood was replaced with sweat to make the game seem less violent, and the attack dogs in the game were replaced by giant mutant rats. Employees of id Software are quoted in The Official DOOM Player Guide about the reaction to Wolfenstein, claiming it to be ironic that it was morally acceptable to shoot people and rats, but not dogs. Two new weapons were added as well. The Super NES version was not as successful as the PC version.

Soundtrack dispute
In May 2020, after the Doom Eternal Original Soundtrack was released, there was a serious backlash to the Doom Eternal OST and accusations of low quality work that did not match composer Mick Gordon's usual standards. On April 19, Gordon confirmed on Twitter that it was not his work, and Marty Stratton subsequently posted on May 20 a 2,500-word open letter on Reddit blaming Gordon for everything that went wrong with the process of creating music for the soundtrack. Following this, public outcry against Gordon reached a level where he received explicit death threats and graphic messages of intent to harm him and his family. Gordon's message accounts, servers, and phones were allegedly inundated with abuse to extreme levels, seriously impacting his mental health.

On November 9, 2022, Mick published a 14,000-word article on Medium explaining his side of the story as a defensive rebuttal of the nine outlined accusations in Stratton's post (described as "an extensive series of lies"), substantiated with various forms of evidence including photographs of emails, receipts, and file metadata to verify his claims. It included claims that Gordon had yet to receive over half of his payment for his work and awards from the soundtrack's nominations at The Game Awards 2020 Stratton had reportedly claimed to deliver on Gordon's behalf; that his name had been listed on the OST's pre-order for weeks before Bethesda had contracted him to work on it just 48 hours before the game's release; Mossholder had been composing an alternate version of the OST as early as August 2019, and in response to request from Gordon's lawyers for Stratton's Reddit post to be removed, Gordon was offered six figures in exchange for a lifetime gag order, but never the possibility of Stratton's defamatory post being removed.

On November 16, 2022, Bethesda released a statement backing Marty Stratton, Chad Mossholder, and everyone in the id software team. Their statement further claimed that they had evidence to rebut Gordon's claims, without releasing mentioned evidence, and expressed concern that his statement enticed harassment and violence towards the team.

People
In 2003, the book Masters of Doom chronicled the development of id Software, concentrating on the personalities and interaction of John Carmack and John Romero. Below are the key people involved with id's success.

John Carmack
Carmack's skill at 3D programming is widely recognized in the software industry and from its inception, he was id's lead programmer. On August 7, 2013, he joined Oculus VR, a company developing virtual reality headsets, and left id Software on November 22, 2013.

John Romero
John Romero saw the horizontal scrolling demo Dangerous Dave in Copyright Infringement and immediately had the idea to form id Software on September 20, 1990. Romero pioneered the game engine licensing business with his "id Summer Seminar" in 1991 where the Keen4 engine was licensed to Apogee for Biomenace. John also worked closely with the DOOM community and was the face of id to its fans. One success of this engagement was the fan-made game Final DOOM, published in 1996. John also created the control scheme for the FPS, and the abstract level design style of DOOM that influenced many 3D games that came after it. John added par times to Wolfenstein 3D, and then DOOM, which started the phenomenon of Speedrunning. Romero wrote almost all the tools that enabled id Software and many others to develop games with id Software's technology. Romero was forced to resign in 1996 after the release of Quake, then later formed the company Ion Storm. There, he became infamous through the development of Daikatana, which was received negatively from reviewers and gamers alike upon release. Afterward, Romero co-founded The Guildhall in Dallas, Texas, served as chairman of the CPL eSports league, created an MMORPG publisher and developer named Gazillion Entertainment, created a hit Facebook game named Ravenwood Fair that garnered 25 million monthly players in 2011, and started Romero Games in Galway, Ireland in 2015.

Both Tom Hall and John Romero have reputations as designers and idea men who have helped shape some of the key PC gaming titles of the 1990s.

Tom Hall
Tom Hall was forced to resign by id Software during the early days of Doom development, but not before he had some impact; for example, he was responsible for the inclusion of teleporters in the game. He was let go before the shareware release of Doom and then went to work for Apogee, developing Rise of the Triad with the "Developers of Incredible Power". When he finished work on that game, he found he was not compatible with the Prey development team at Apogee, and therefore left to join his ex-id Software compatriot John Romero at Ion Storm. Hall has frequently commented that if he could obtain the rights to Commander Keen, he would immediately develop another Keen title.

Sandy Petersen
Sandy Petersen was a level designer for 19 of the 27 levels in the original Doom title as well as 17 of the 32 levels of Doom II. As a fan of H.P. Lovecraft, his influence is apparent in the Lovecraftian feel of the monsters for Quake, and he created Inferno, the third "episode" of the first Doom. He was forced to resign from id Software during the production of Quake II and most of his work was scrapped before the title was released.

American McGee
American McGee was a level designer for Doom II, The Ultimate Doom, Quake, and Quake II. He was asked to resign after the release of Quake II, and he then moved to Electronic Arts where he gained industry notoriety with the development of his own game American McGee's Alice. After leaving Electronic Arts, he became an independent entrepreneur and game developer. McGee headed the independent game development studio Spicy Horse in Shanghai, China from 2007 to 2016.

title: Id Software


text:
Isaac Stern

Isaac Stern (July 21, 1920 – September 22, 2001) was an American violinist.

Born in Ukraine, Stern moved to the US when he was 14 months old. Stern performed both nationally and internationally, notably touring the Soviet Union and China, and performing extensively in Israel, a country to which he had close ties since shortly after its founding.

Stern received extensive recognition for his work, including winning the Presidential Medal of Freedom and six Grammy Awards, and being named to the French Legion of Honour. The Isaac Stern Auditorium at Carnegie Hall bears his name, due to his role in saving the venue from demolition in the 1960s.

Biography
The son of Solomon and Clara Stern, Isaac Stern was born in Kremenets, Poland (now Ukraine), into a Jewish family. He was 14 months old when his family moved to San Francisco in 1921. Both his parents were musical, and his mother, who had studied at the Saint Petersburg Conservatory, began teaching him the piano when he was six, before switching to the violin when he was eight. In 1928, Stern’s parents enrolled him at the San Francisco Conservatory of Music, where he studied until 1931 before going on to study briefly in New York with Louis Persinger. He returned to the San Francisco Conservatory to study for five years with Naoum Blinder, the concertmaster of the San Francisco Symphony, to whom he said he owed the most. At his public début on February 18, 1936, aged 15, he played Saint-Saëns' Violin Concerto No. 3 in B minor with the San Francisco Symphony under the direction of Pierre Monteux. Reflecting on his background, Stern once memorably quipped that cultural exchanges between the U.S. and Soviet Russia drew from the same city:

 * "They send us their Jews from Odessa, and we send them our Jews from Odessa."

During World War II, Stern was rejected from military service due to flat feet. He then joined the United Service Organizations and performed for US troops. During one such performance on Guadalcanal, a Japanese soldier, mesmerized by his playing, sneaked into the audience of US personnel listening to his performance before sneaking back out.

Stern toured the Soviet Union in 1951, the first American violinist to do so. In 1967, Stern stated his refusal to return to the USSR until the Soviet regime allowed artists to enter and leave the country freely. His only visit to Germany was in 1999, for a series of master classes, but he never performed publicly in Germany.

Stern was married three times. His first marriage, in 1948 to ballerina Nora Kaye, ended in divorce after 18 months, but the two of them remained friends. On August 17, 1951, he married Vera Lindenblit (1927–2015). They had three children together, including conductors Michael and David Stern and also Rabbi Shira Stern, one of the first female rabbis in the USA. Their marriage ended in divorce in 1994 after 43 years. In 1996, Stern married his third wife, Linda Reynolds. His third wife, his three children, and his five grandchildren survived him.

Stern died September 22, 2001, of heart failure in a Manhattan, New York, hospital after an extended stay.

Music career
In 1940, Stern began performing with Russian-born pianist Alexander Zakin, collaborating until 1977. Within musical circles, Stern became renowned both for his recordings and for championing certain younger players. Among his discoveries were cellists Yo-Yo Ma and Jian Wang, and violinists Itzhak Perlman and Pinchas Zukerman.

In the 1960s, he played a major role in saving New York City's Carnegie Hall from demolition, by organising the Citizens' Committee to Save Carnegie Hall. Following the purchase of Carnegie Hall by New York City, the Carnegie Hall Corporation was formed, and Stern was chosen as its first president, a title he held until his death. Carnegie Hall later named its main auditorium in his honor.

Among Stern's many recordings are concertos by Brahms, Bach, Beethoven, Mendelssohn, Sibelius, Tchaikovsky, and Vivaldi and modern works by Barber, Bartók, Stravinsky, Bernstein, Rochberg, and Dutilleux. The Dutilleux concerto, entitled L'arbre des songes ["The Tree of Dreams"] was a 1985 commission by Stern himself. He also dubbed actors' violin-playing in several films, such as Fiddler on the Roof.

Stern served as musical advisor for the 1946 film, Humoresque, about a rising violin star and his patron, played respectively by John Garfield and Joan Crawford. He was also the featured violin soloist on the soundtrack for the 1971 film of Fiddler on the Roof. In 1999, he appeared in the film Music of the Heart, along with Itzhak Perlman and several other famed violinists, with a youth orchestra led by Meryl Streep (the film was based on the true story of a gifted violin teacher in Harlem who eventually took her musicians to play a concert in Carnegie Hall).

In his autobiography, co-authored with Chaim Potok, My First 79 Years, Stern cited Nathan Milstein and Arthur Grumiaux as major influences on his style of playing.

He won Grammys for his work with Eugene Istomin and Leonard Rose in their famous chamber music trio in the 1960s and '70s, while also continuing his duo work with Alexander Zakin during this time. Stern recorded a series of piano quartets in the 1980s and 1990s with Emanuel Ax, Jaime Laredo and Yo-Yo Ma, including those of Mozart, Beethoven, Schumann and Fauré, winning another Grammy in 1992 for the Brahms quartets Opp. 25 and 26.

In 1979, seven years after Richard Nixon made the first official visit by a US president to the country, the People's Republic of China offered Stern and pianist David Golub an unprecedented invitation to tour the country. While there, he collaborated with the China Central Symphony Society (now China National Symphony) under the direction of conductor Li Delun. Their visit was filmed and resulted in the Oscar-winning documentary, From Mao to Mozart: Isaac Stern in China.

Ties to Israel
Stern maintained close ties with Israel. Stern began performing in the country in 1949. In 1973, he performed for wounded Israeli soldiers during the Yom Kippur War. During the 1991 Gulf War and Iraq's Scud missile attacks on Israel, he had been playing in the Jerusalem Theater. During his performance, an air raid siren sounded, causing the audience to panic. Stern then stepped onto the stage and began playing a movement of Bach. The audience then calmed down, donned gas masks, and sat throughout the rest of his performance. Stern was a supporter of several educational projects in Israel, among them the America-Israel Foundation and the Jerusalem Music Center.

Instruments
Stern's favorite instrument was the Ysaÿe Guarnerius, one of the violins produced by the Cremonese luthier Giuseppe Guarneri del Gesù. It had previously been played by the violin virtuoso and composer Eugène Ysaÿe.

Among other instruments, Stern played the "Kruse-Vormbaum" Stradivarius (1728), the "ex-Stern" Bergonzi (1733), the "Panette" Guarneri del Gesù (1737), a Michele Angelo Bergonzi (1739–1757), the "Arma Senkrah" Guadagnini (1750), a Giovanni Guadagnini (1754), a J. B. Vuillaume copy of the "Panette" Guarneri del Gesu of 1737 (c.1850), and the "ex-Nicolas I" J.B. Vuillaume (1840). He also owned two contemporary instruments by Samuel Zygmuntowicz and modern Italian Jago Peternella Violins.

In May 2003, Stern's collection of instruments, bows and musical ephemera was sold through Tarisio Auctions. The auction set a number of world records and was at the time the second highest grossing violin auction of all time, with total sales of over $3.3M.

Awards and commemoration

 * Sonning Award (1982; Denmark)
 * Wolf Prize
 * Kennedy Center Honors (1984)
 * Grammy Award for Best Instrumental Soloist(s) Performance (with orchestra) (1962, 1963, 1965, 1982)
 * Grammy Award for Best Chamber Music Performance (1971, 1992)
 * National Medal of Arts (1991)
 * Presidential Medal of Freedom (1992)
 * Elected to the American Philosophical Society (1995)
 * Polar Music Prize (2000; Sweden)
 * Commandeur de la Légion d'honneur (1990)
 * Gold Medal of the Royal Philharmonic Society (1991)
 * Carnegie Hall Midtown Manhattan, New York: main auditorium was named for Isaac Stern in 1997.

In 2012, a street in Tel Aviv was named for Stern.

Discography

 * 1944
 * Brahms: String Sextet No. 1 (with Alexander Schneider, Milton Katims, Milton Thomas, Pablo Casals and Madeleine Foley)

 * 1944
 * Brahms: Trio for Piano, Violin and Cello No. 1 in B Major, op. 8 (with Myra Hess and Pablo Casals)

 * 1946
 * Violin selections from the movie Humoresque (1946 film) with Oscar Levant on the piano, Columbia Masterworks Records set MM-657

 * 1951
 * Brahms: Violin Concerto in D Major (with Sir Thomas Beecham and the Royal Philharmonic Orchestra), Columbia Records

 * 1952
 * Bach: Partita in E Minor & G Minor for Violin and Piano, Sonata No.3 in E Major for Violin and Piano (with Alexander Zakin)

 * 1953
 * Prokofiev: Violin Sonatas (with Alexander Zakin)

 * 1957
 * Wieniawski: Violin Concerto No. 2 in D Minor, op. 22 (with Philadelphia Orchestra; conductor: Eugene Ormandy)

 * 1958
 * Tchaikovsky: Violin Concerto in D Major op. 35 (with Philadelphia Orchestra; conductor: Eugene Ormandy)
 * Mendelssohn: Violin Concerto in e minor op. 64 (with Philadelphia Orchestra; conductor: Eugene Ormandy)

 * 1959
 * Saint-Saens: Introduction & Rondo Capriccioso op. 28 (with Philadelphia Orchestra; conductor: Eugene Ormandy)
 * Beethoven Violin Concerto in D, Op. 61 (with New York Philharmonic; conductor: Leonard Bernstein) Fontana 699 049 CL CFL 1051.

 * 1964
 * Hindemith: Violin Concerto (1939) (with New York Philharmonic; conductor: Leonard Bernstein)

 * 1978
 * Penderecki: Violin Concerto No. 1 (1976)(with Minnesota Orchestra; conductor: Stanislaw Skrowaczewski)

 * 1983
 * Bach, Vivaldi: Concertos for 2 Violins
 * Isaac Stern: 60th Anniversary Celebration
 * Mendelssohn: Violin Concerto; Beethoven: Romances in G & F Major
 * Haydn: London Trios

 * 1984
 * Barber Violin Concerto

 * 1985
 * An Isaac Stern Vivaldi Gala

 * 1986
 * Tchaikovsky, Mendelssohn: Violin Concertos

 * 1987
 * Dutilleux: L'Arbre des Songes (Concerto pour Violin et Orchestre)
 * Maxwell Davies: Concerto for Violin and Orchestra
 * Celebration
 * Bach: Double Concerto; Violin Concertos Nos.1 & 2
 * Beethoven: Violin Concerto
 * Mozart: The Flute Quartets
 * Bach: Concertos for Violin, BWV 1041–43 & 1060

 * 1988
 * Shostakovich: Piano Trio No.2; Cello Sonata
 * Brahms: Concerto for Violin, Cello and Orchestra in A Minor, Op. 102 & Piano Quartet No. 3 in C Minor, Op. 60
 * Prokofiev: Violin Concertos No. 1 & 2
 * Brahms: Violin Concerto

 * 1989
 * The Japanese Album
 * Music, My Love
 * Prokofiev: Concertos No. 1 & 2 for Violin and Orchestra
 * Mozart: Violin Concertos Nos.4 & 5

 * 1990
 * Brahms, Mendelssohn, Schubert: Trios
 * Brahms: The Piano Quartets
 * Rameau: Pieces de clavecin en concerts
 * Lalo, Bruch, Wenianski, others: Violin Concertos
 * Bach, Mozart, Brahms, others: Violin Concertos
 * Mozart, Telemann, J.C. Bach, Reicha: Trios, Quartets
 * Schubert: Violin Sonatas
 * Humoresque: Favorite Violin Encores

 * 1991
 * Beethoven: Piano Concerto No.5 "Emperor"; Triple Concerto
 * Beethoven: Complete Trios
 * Concert of the Century: Celebrating the 85th Anniversary of Carnegie Hall
 * Dvorák: Cello Concerto; Violin Concerto
 * Webern: Complete Works, Op. 1 – Op. 31

 * 1992
 * Brahms: Sextets; more
 * Beethoven & Schumann Piano Quartets (with Emanuel Ax, Jaime Loredo, & Yo-Yo Ma)

 * 1993
 * Tchaikovsky: Concerto for Violin and Orchestra & Serenade for Strings
 * Fauré: Piano Quartets

 * 1994
 * Greatest Hits: Violin
 * The House of Magical Sounds
 * Greatest Hits: Schubert
 * Greatest Hits: Brahms
 * Beethoven, Schumann: Piano Quartets
 * Mozart: Sonatas for Violin and Piano, K. 454, 296 & 526
 * Beethoven: Piano Trios "Ghost" & "Archduke"
 * Bach: Violin Concerto, BWV 1041; Piano Concerto, BWV 1056; Brandenburg Concerto No.5; more
 * Mozart: Sinfonia Concertante; Violin Concerto No.5
 * Brahms: Sextet in B-flat major, Op. 18 & Piano Trio No. 1 in B major, Op. 8
 * Schubert: Quintet in C major, D956 & Symphony No. 5 in B-flat major, D485

 * 1995
 * Isaac Stern Presents Encores with Orchestra
 * Telemann, Bach Family: Trio Sonatas
 * Mendelssohn: Piano Trios 1 & 2
 * Brahms: Piano Trios, Piano Quartets
 * A Life in Music, Vol.3: Bach, Beethoven, Brahms, Mozart, more
 * Beethoven: Piano Trios "Ghost" & "Archduke"; Variations
 * Schubert, Haydn: Piano Trios; Mozart: Piano Quartet
 * Bartók: Violin Concertos
 * Bernstein/Dutilleux: Violin Concertos
 * Berg: Violin Concerto; Kammerkonzert
 * Prokofiev/Bartók: Violin Concertos; Rhapsody No.1
 * Stravinsky/Rochberg: Violin Concertos
 * Barber/Maxwell Davies: Violin Concertos
 * Hindemith/Penderecki: Violin Concertos
 * Berg: Piano Sonata; Krenek: Piano Sonata No.3; Webern: Piano Variations; Debussy, Ravel: works
 * A Life in Music, Vol.1: Beethoven, Brahms, Mendelssohn, Sibelius, more
 * Mozart: Haffner Serenade
 * Mozart: Sonatas for Violin and Piano, Vol. II
 * Beethoven, Brahms: Violin Concertos
 * Tchaikovsky/Sibelius: Violin Concertos
 * Bach: Violin Concertos; Double Concerto; more
 * Vivaldi: The Four Seasons; Concertos
 * Mozart: Violin Concertos Nos.1–5; Sinfonia concertante; more
 * Wieniawski/Bruch/Tchaikovsky: Violin Concertos
 * Mendelssohn/Dvorák: Violin Concertos
 * Saint-Saëns: Violin concerto n°3, Lalo: Symphonie Espagnole, Chausson: Poème, Fauré: Berceuse, Ravel: Tzigane

 * 1996
 * More Mozart's Greatest Hits
 * Mozart: Violin Sonatas, Vol. III
 * Schubert and Boccherini String Quintets
 * A Life in Music, Vol.4: Bach, Bartók, Beethoven, Copland, Schubert, more
 * Prokofiev: Violin Sonatas
 * Bartók: Violin Sonatas; Webern: Four Pieces for Violin and Piano
 * Beethoven: Violin Sonatas
 * J.S. & C.P.E. Bach, Handel, Tartini: Violin Sonatas
 * Hindemith/Bloch/Copland: Violin Sonatas
 * Schubert: Sonatinas Nos.1–3; Rondeau Brillant; Grand Duo Sonata
 * Franck/Debussy/Enesco: Violin Sonatas
 * Brahms: Violin Sonatas No. 1-3
 * Isaac Stern Presents Encores with Violin & Piano

 * 1997
 * Barber: Adagio for Strings / Schuman – In Praise of Shahn etc.
 * Bartók Sonatas for Violin and Piano
 * Mozart: The Piano Quartets

 * 1998
 * Isaac Stern Plays Mozart, Beethoven, Haydn
 * Beethoven: Violin Concerto in D
 * Bernstein: The Age of Anxiety; Foss: Serenade
 * Bach, Vivaldi: Concertos
 * Caprice Viennois: Music of Kreisler

 * 1999
 * My First 79 Years
 * Tchaikovsky, Mendelssohn: Violin Concertos

 * 2000
 * Dvorák: Piano Quartet No.2, Sonatina in G, Romantic Pieces
 * Vivaldi: The Four Seasons; Concertos for Two Violins

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Isaac Stern


text:
Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.

A definite integral computes 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 also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.

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 general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.

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 two points in space. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

Pre-calculus integration
The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus and philosopher Democritus (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.

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.

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 AD) derived a formula for the sum of fourth powers. Alhazen determined the equations to calculate the area enclosed by the curve represented by $$y=x^k$$ (which translates to the integral $$\int x^k \, dx$$ in contemporary notation), for any given non-negative integer value of $$k$$. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.

The 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 $x^{n}$ up to degree $n = 9$ in Cavalieri's quadrature formula. The case n = −1 required the invention of a function, the hyperbolic logarithm, achieved by quadrature of the hyperbola in 1647.

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 $x$ to a general power, including negative powers and fractional powers.

Leibniz and Newton
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 with continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.

Formalization
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.

Historical notation
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 1819–1820, reprinted in his book of 1822.

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 $. x$ or $x′$, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

First use of the term
The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur".

Terminology and notation
In general, the integral of a real-valued function $f(x)$ with respect to a real variable $x$ on an interval $[a, b]$ is written as
 * $$\int_{a}^{b} f(x) \,dx.$$

The integral sign $∫$ represents integration. The symbol $dx$, called the differential of the variable $x$, indicates that the variable of integration is $x$. The function $f(x)$ is called the integrand, the points $a$ and $b$ are called the limits (or bounds) of integration, and the integral is said to be over the interval $[a, b]$, called the interval of integration. 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.

When the limits are omitted, as in

 * $$\int f(x) \,dx,$$

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).

In advanced settings, it is not uncommon to leave out $dx$ when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write $\int_a^b (c_1f+c_2g) = c_1\int_a^b f + c_2\int_a^b g $ to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.

Interpretations
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.

As another example, to find the area of the region bounded by the graph of the function $\sqrt{x}$ $\sqrt{x}$ between $f(x) =$ and $x = 0$, one can divide the interval into five pieces ($x = 1$), then construct rectangles using the right end height of each piece (thus $0, 1/5, 2/5, ..., 1$) and sum their areas to get the approximation

 * $$\textstyle \sqrt{\frac{1}{5}}\left(\frac{1}{5}-0\right)+\sqrt{\frac{2}{5}}\left(\frac{2}{5}-\frac{1}{5}\right)+\cdots+\sqrt{\frac{5}{5}}\left(\frac{5}{5}-\frac{4}{5}\right)\approx 0.7497,$$

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 increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, $\sqrt{0}, \sqrt{1/5}, \sqrt{2/5}, ..., \sqrt{1}$). One writes

 * $$\int_{0}^{1} \sqrt{x} \,dx = \frac{2}{3},$$

which means $2/3$ is the result of a weighted sum of function values, $2/3$, multiplied by the infinitesimal step widths, denoted by $\sqrt{x}$, on the interval $dx$.

Formal definitions
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 are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.

Riemann integral
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 $[0, 1]$ on the real line is a finite sequence

 * $$ a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$$

This partitions the interval $y = x^{2}$ into $n$ sub-intervals $y = x^{2}$ indexed by $i$, each of which is "tagged" with a specific point $[a, b]$. A Riemann sum of a function $f$ with respect to such a tagged partition is defined as

 * $$\sum_{i=1}^n f(t_i) \, \Delta_i ; $$

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, $[a, b]$. The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, $[x_{i−1}, x_{i}]$. The Riemann integral of a function $f$ over the interval $t_{i} ∈ [x_{i−1}, x_{i}]$ is equal to $S$ if:

 * For all $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that, for any tagged partition $$[a, b]$$ with mesh less than $$\delta$$,

 * $$\left| S - \sum_{i=1}^n f(t_i) \, \Delta_i \right| < \varepsilon.$$

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

Lebesgue integral
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.

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:

"I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral."

As Folland puts it, "To compute the Riemann integral of $f$, one partitions the domain $[a, b]$ into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of $f$ ". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure $Δ_{i} = x_{i}−x_{i−1}$ of an interval $max_{i=1...n} Δ_{i}$ is its width, $[a, b]$, 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.

Using the "partitioning the range of $f$ " philosophy, the integral of a non-negative function $μ(A)$ should be the sum over $t$ of the areas between a thin horizontal strip between $A = [a, b]$ and $b − a$. This area is just $f : R → R$. Let $y = t$. The Lebesgue integral of $f$ is then defined by

 * $$\int f = \int_0^\infty f^*(t)\,dt$$

where the integral on the right is an ordinary improper Riemann integral ($y = t + dt$ 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.

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

 * $$\int_E |f|\,d\mu < + \infty.$$

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

 * $$\int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu$$

where

 * $$\begin{alignat}{3}

& f^+(x) &&{}={} \max \{f(x),0\} &&{}={} \begin{cases} f(x), & \text{if } f(x) > 0, \\ 0, & \text{otherwise,} \end{cases}\\ & f^-(x) &&{}={} \max \{-f(x),0\} &&{}={} \begin{cases} -f(x), & \text{if } f(x) < 0, \\ 0, & \text{otherwise.} \end{cases} \end{alignat}$$

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

 * The Darboux integral, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.
 * The Riemann–Stieltjes integral, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
 * The Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals.
 * The Daniell integral, which subsumes the Lebesgue integral and Lebesgue–Stieltjes integral without depending on measures.
 * The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.
 * The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
 * The Itô integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
 * The Young integral, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation.
 * The rough path integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
 * The Choquet integral, a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
 * The Bochner integral, an extension of the Lebesgue integral to a more general class of functions, namely, those with a domain that is a Banach space.

Linearity
The collection of Riemann-integrable functions on a closed interval $μ{ x : f(x) > t} dt$ forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

 * $$ f \mapsto \int_a^b f(x) \; dx$$

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:

 * $$ \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,$$

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

 * $$ f\mapsto \int_E f \, d\mu $$

is a linear functional on this vector space, so that:

 * $$ \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. $$

More generally, consider the vector space of all measurable functions on a measure space $f^{∗}(t) = μ{ x : f(x) > t }$, taking values in a locally compact complete topological vector space $V$ over a locally compact topological field $f$. Then one may define an abstract integration map assigning to each function $f$ an element of $V$ or the symbol $[a, b]$,

 * $$ f\mapsto\int_E f \,d\mu, \,$$

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

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 $X$, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See for an axiomatic characterization of the integral.

Inequalities
A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval $[a, b]$ and can be generalized to other notions of integral (Lebesgue and Daniell).

(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.$$ If $f$ is Riemann-integrable on $[a, b]$ then the same is true for $C$, and $$\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. $$ Moreover, if $m$ and $M$ are both Riemann-integrable then $Q_{p}$ is also Riemann-integrable, and $$\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). $$ This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions $x$ and $[a, b]$ on the interval $f$. \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.$$ For $K = C$, Hölder's inequality becomes the Cauchy–Schwarz inequality. \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} + \left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.$$ An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.
 * Upper and lower bounds. An integrable function $[a, b]$ on $x$, is necessarily bounded on that interval. Thus there are real numbers $[a, b]$ and $f$ so that $m ≤ f (x) ≤ M$ for all $g$ in $M$. Since the lower and upper sums of $[a, b]$ over $x$ are therefore bounded by, respectively, $m(b − a)$ and $M(b − a)$, it follows that $$ m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). $$
 * Inequalities between functions. If $f(x) ≤ g(x)$ for each $[a, b]$ in $[c, d]$ then each of the upper and lower sums of $[a, b]$ is bounded above by the upper and lower sums, respectively, of $x$. Thus $$ \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. $$ This is a generalization of the above inequalities, as $M(b − a)$ is the integral of the constant function with value $f$ over $g$. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if $f(x) < g(x)$ for each $f$ in $[a, b]$, then $$ \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. $$
 * Subintervals. If $f$ is a subinterval of $g$ and $f (x)$ is non-negative for all $f$, then $$ \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. $$
 * Products and absolute values of functions. If $g$ and $[a, b]$ are two functions, then we may consider their pointwise products and powers, and absolute values: $$
 * Hölder's inequality. Suppose that $p$ and $q$ are two real numbers, $|f|$ with $fg$, and $f$ and $g$ are two Riemann-integrable functions. Then the functions $1 ≤ p, q ≤ ∞$ and $1⁄p + 1⁄q = 1$ are also integrable and the following Hölder's inequality holds: $$\left|\int f(x)g(x)\,dx\right| \leq
 * Minkowski inequality. Suppose that $|f|^{p}$ is a real number and $f$ and $g$ are Riemann-integrable functions. Then $|g|^{q}$ and $p = q = 2$ are also Riemann-integrable and the following Minkowski inequality holds: $$\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq

Conventions
In this section, $f$ is a real-valued Riemann-integrable function. The integral

 * $$ \int_a^b f(x) \, dx $$

over an interval $p ≥ 1$ is defined if $| f |^{p}, | g |^{p}$. This means that the upper and lower sums of the function $f$ are evaluated on a partition $| f + g |^{p}$ whose values $[a, b]$ are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating $f$ within intervals $a < b$ where an interval with a higher index lies to the right of one with a lower index. The values $a$ and $b$, the end-points of the interval, are called the limits of integration of $f$. Integrals can also be defined if $a = x_{0} ≤ x_{1} ≤. . . ≤ x_{n} = b$:

 * $$\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. $$

With $x_{i}$, this implies:

 * $$\int_a^a f(x) \, dx = 0. $$

The first convention is necessary in consideration of taking integrals over subintervals of $[x_{ i}, x_{ i +1}]$; 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 $f$ on an interval $a > b$ implies that $f$ is integrable on any subinterval $a = b$, but in particular integrals have the property that if $c$ is any element of $[a, b]$, then:

 * $$ \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$$

With the first convention, the resulting relation

 * $$\begin{align}

\int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\ &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end{align}$$

is then well-defined for any cyclic permutation of $a$, $b$, and $c$.

Fundamental theorem of calculus
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.

First theorem
Let $f$ be a continuous real-valued function defined on a closed interval $[a, b]$. Let $F$ be the function defined, for all $x$ in $[c, d]$, by

 * $$F(x) = \int_a^x f(t)\, dt.$$

Then, $F$ is continuous on $[a, b]$, differentiable on the open interval $[a, b]$, and

 * $$F'(x) = f(x)$$

for all $x$ in $[a, b]$.

Second theorem
Let $f$ be a real-valued function defined on a closed interval [$[a, b]$] that admits an antiderivative $F$ on $(a, b)$. That is, $f$ and $F$ are functions such that for all $x$ in $(a, b)$,

 * $$f(x) = F'(x).$$

If $f$ is integrable on $a, b$ then

 * $$\int_a^b f(x)\,dx = F(b) - F(a).$$

Improper integrals
Improper_integral.svg$$\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi$$

has unbounded intervals for both domain and range.]] 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.

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

 * $$\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx.$$

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

 * $$\int_a^b f(x)\,dx = \lim_{\varepsilon \to 0} \int_{a+\epsilon}^{b} f(x)\,dx.$$

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 $[a, b]$, or $[a, b]$. In more complicated cases, limits are required at both endpoints, or at interior points.

Multiple integration
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 $$R=[a,b]\times [c,d]$$ can be written

 * $$\int_R f(x,y)\,dA$$

where the differential $( a, b ]$ 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

 * $$\int_a^b\left[\int_c^d f(x,y)\,dy\right]\,dx.$$

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:

 * $$\iint_R f(x,y) \, dA.$$

Integration over more general domains is possible. The integral of a function f, with respect to volume, over an n-dimensional region D of $$\mathbb{R}^n$$ is denoted by symbols such as:

 * $$\int_D f(\mathbf x) d^n\mathbf x \ = \int_D f\,dV.$$

Line integrals and surface integrals
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.

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.

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, $dA$, may be expressed (in terms of vector quantities) as:

 * $$W=\mathbf F\cdot\mathbf s.$$

For an object moving along a path $C$ in a vector field $z = f(x,y)$ 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 $F$ to $s$. This gives the line integral

 * $$W=\int_C \mathbf F\cdot d\mathbf s.$$

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.

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

 * $$\int_S {\mathbf v}\cdot \,d{\mathbf S}.$$

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.

Contour integrals
In complex analysis, the integrand is a complex-valued function of a complex variable $z$ instead of a real function of a real variable $x$. When a complex function is integrated along a curve $$\gamma$$ in the complex plane, the integral is denoted as follows

 * $$\int_\gamma f(z)\,dz.$$

This is known as a contour integral.

Integrals of differential forms
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:

 * $$E(x,y,z)\,dx + F(x,y,z)\,dy + G(x,y,z)\, dz$$

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.

A differential two-form is a sum of the form

 * $$G(x,y,z) \, dx\wedge dy + E(x,y,z) \, dy\wedge dz + F(x,y,z) \, dz\wedge dx.$$

Here the basic two-forms $$dx\wedge dy, dz\wedge dx, dy\wedge dz$$ measure oriented areas parallel to the coordinate two-planes. The symbol $$\wedge$$ 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 $$E\mathbf i+F\mathbf j+G\mathbf k$$.

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.

Summations
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.

Functional integrals
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.

Applications
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.

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, $$\pi r^2 h $$, where $$r$$ is the radius. In the case of a simple disc created by rotating a curve about the $S$-axis, the radius is given by $v$, and its height is the differential $S$. Using an integral with bounds $x$ and $f(x)$, the volume of the disc is equal to: $$\pi \int_a^b f^2 (x) \, dx.$$Integrals 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 $$[a,b]$$ is given by
 * $$x(b)-x(a) = \int_a^b v(t) \,dt,$$

where $$v(t)$$ is the velocity expressed as a function of time. The work done by a force $$F(x)$$ (given as a function of position) from an initial position $$A$$ to a final position $$B$$ is:

 * $$W_{A\rightarrow B} = \int_A^B F(x)\,dx.$$

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

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

 * $$\int_a^b f(x)\,dx=F(b)-F(a).$$

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.

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.

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

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

Symbolic
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.

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 the integral if is elementary. 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.

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 Risch's algorithm to include such functions is possible but challenging and has been an active research subject.

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.

Rule-based integration systems facilitate integration. Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands. This system uses over 6600 integration rules to compute integrals. The method of brackets is a generalization of Ramanujan's master theorem that can be applied to a wide range of univariate and multivariate integrals. A set of rules are applied to the coefficients and exponential terms of the integrand's power series expansion to determine the integral. The method is closely related to the Mellin transform.

Numerical
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.

Riemann sums, the trapezoidal rule, and Simpson's rule are examples of a family of quadrature rules called the Newton–Cotes formulas. The degree $n$ Newton–Cotes quadrature rule approximates the polynomial on each subinterval by a degree $n$ 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.

Romberg's method halves the step widths incrementally, giving trapezoid approximations denoted by $f(x)$, $[a, b]$, and so on, where $F′ = f$ is half of $T(h_{0})$. 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 $T(h_{1})$. Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials. An $n$-point Gaussian method is exact for polynomials of degree up to $h_{k+1}$.

The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.

Mechanical
The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called planimeter. The volume of irregular objects can be measured with precision by the fluid displaced as the object is submerged.

Geometrical
Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

Integration by differentiation
Kempf, Jackson and Morales demonstrated mathematical relations that allow an integral to be calculated by means of differentiation. Their calculus involves the Dirac delta function and the partial derivative operator $$\partial_x$$. This can also be applied to functional integrals, allowing them to be computed by functional differentiation.

Using the fundamental theorem of calculus
The fundamental theorem of calculus allows straightforward calculations of basic functions:

 * $$\int_0^\pi \sin(x) \,dx = -\cos(x) \big|^{x = \pi}_{x = 0} = -\cos(\pi) - \big(-\cos(0)\big) = 2.$$

Online books

 * Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
 * Stroyan, K. D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
 * Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
 * Crowell, Benjamin, Calculus, Fullerton College, an online textbook
 * Garrett, Paul, Notes on First-Year Calculus
 * Hussain, Faraz, Understanding Calculus, an online textbook
 * Johnson, William Woolsey (1909) Elementary Treatise on Integral Calculus, link from HathiTrust.
 * Kowalk, W. P., Integration Theory, University of Oldenburg. A new concept to an old problem. Online textbook
 * Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
 * Numerical Methods of Integration at Holistic Numerical Methods Institute
 * P. S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) — a cookbook of definite integral techniques

title: Integral