Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval, then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

Through history, Metz population has been impacted by the vicissitudes of the wars and the annexations involving the city and avoiding a continuous population growth.

Through maneuver and continuous assault, Chinese, Persian, Arab and Eastern European armies could be stressed until they broke, and then were annihilated in pursuit.

Through daily efforts to rebuild, re-arm and improve the fortifications, he was able to finally connect them with a continuous defence system enceinte.

Through touring heavily throughout 1999 and 2000, Guided by Voices ' live act became legendary, with shows often stretching past the three-hour mark, and populated by an endless stream of new and classic songs, Pollard solo tracks, impromptu covers of The Who, David Bowie and The Rolling Stones, all accompanied by continuous alcohol consumption.

Through September 28, for example, Alabama ’ s 117th Air Refueling Wing kept aircraft aloft on a continuous basis.

Through innuendo, rumors, and public discrediting, a hostile environment is created in which one individual gathers others to willingly, or unwillingly, participate in continuous malevolent actions to force a person out of the workplace.

Through this strategy, 40 million bank and investment accounts were quietly converted into war bonds, providing the Reich government with a continuous supply of money.

Through the continuous cycle of learning, teacher ’ s often work with students to develop a framework of knowledge, which is to be evaluated based on student input to the lessons content.

Through unusual animation techniques the illusion is created that the film consists of one continuous shot or long take.

Through his continuous propagation of modern art as shown in his works and teachings, Edades proved that modernists were not fooling people as Guillermo Tolentino asserted.

Through continuous research, their work is to identify and translate documents and texts that shed new insight into this important period of history.

Through this monitoring system, the concept of continuous self-upgrading and development after a training course can be impacted on all teachers in Rizal Province.

Through this area, it had a continuous center turn lane, losing the grassy median it had in places downtown.

His second book was The Man Who Walked Through Time ( 1968 ), in which Fletcher was the first person to walk a continuous route through Grand Canyon National Park in 1963.

Through the establishment of the SERC, the University seeks to help and support the local software industry in establishing and improving their processes and practices through continuous feedback and training.

Through the House of Burgesses, the Virginia House of Delegates is considered the oldest continuous legislative body in the New World.

" Through continuous struggle the Afghan Mujahideen succeeded in their mission, and the Russian forces withdrew from Afghanistan in 1989 after the loss of tens of thousands of its soldiers.

Through its comprehensive product range and its reactivity to market innovations, Olympus enjoys a virtual stranglehold of the world market in gastro-intestinal endoscopes.

Through the 1930s and 1940s sportfishermen in Florida, amongst them John Rybovich and Ernest Hemingway, continued to innovate and refine, and in 1946 the Rybovich yard launched the Miss Chevy II, a 34-footer that crystallized all the innovations that had gone before into a design whose features-raised foredeck, flybridge controls and roomy cockpit-are still closely followed by today's leading sportfish builders.

Through innovations from Hector DeLuca, Vitamin D continues to be a large percentage of WARF's income, around 70 %.

Through education, research, and engagement, EDGE identifies the most promising pathways toward a sustainable energy system and economy, works with leading firms to drive these innovations across their value chains, and helps seed the partnerships that can accelerate these transformations.

Through the accident of his exile at Paris he was thrown into relations with the leaders of the French Romantic Movement, and was so far impressed with the innovations of the new school as to write in French a romantic piece entitled Abén Humeya ( 1830 ), which was played at the Porte Saint-Martin.

Through further insertion of non-day units of time, the calendar year can be further synchronized ; however, these extra units of time are not considered part of the calendar.

Through volcanic activity due to thermal expansion this continent broke and the new continents drifted away from each other because of further expansion of the rip-zones, where the oceans now lie.

Through Vrtanes, Gregory and Miriam would have further descendants and when Gregory died, Aristaces succeeded him.

Through the authority vested in ` Abdu ' l-Bahá in the Aqdas there is an expanse of internationalism related to the law in works like The Secret of Divine Civilization and through his extended authority to Shoghi Effendi works like his World Order of Bahá ' u ' lláh further elaborates on the internationalism theme.

Through the impulse given by the Catholic monarchs like Theodelinda, Liutprand, and Desiderius to the foundation of monasteries to further their political control, Lombard architecture flourished.

Through the afternoon, Soult, Augereau, and Murat managed to hold their ground while Davout, assisted by Saint-Hilaire, gradually bent the Russian left back further and further.

Through further developments, it was found the waterjet offered several other advantages over propellers for a wide range of vessel types, and as such, waterjets are used widely today for many high-speed vessels, including passenger ferries, rescue craft, patrol boats and offshore supply vessels.

Through a studio known as the London Suffrage Atelier, she contributed artwork to further the cause of women's suffrage in Great Britain.

Through the years he dabbled in writing and directing ; however, he never pursued either form further.

Through this recognition came further evolution.

Through her mother, the Princess is a descendant of Diane de Poitiers, mistress of Henry II of France ,< ref > And to add further, great < sup > 16 </ sup >- granddaughter of Agnès Sorel ; Diane's husband is Agnès ' grandson thru her daughter Charlotte, Fille de France.

Through section 2 of the Canada Act 1982, the United Kingdom ended its involvement with further amendments to the Canadian constitution.

Historian Avril Cameron explains what the Council of Chalcedon meant in greater detail, " It developed and clarified the creed of Nicaea, according to which God was Father, Son and Holy Spirit, by further proclaiming that Christ was at all times after the Incarnation fully God and fully human ..." Through both councils, Pulcheria devoted the last years of her life to the idea that she was the Theotokos, and had three churches in Constantinople dedicated to the Virgin Mary.

Through the years, the park was developed further with the addition of Quaker Lake, winterized cabins and full service cottages, hiking and snowmobile trails, picnic and recreation areas, bike and horse paths, and better roads enabling people to access the park.

Through sensationalism, he claims, the audience was further educated and encouraged to take more interest in the news.

Through the Sakura Project, the Japanese Embassy donated a further 34 cherry trees to High Park in 2001, plus cherry trees to various other locations like Exhibition Place, McMaster University, York University and the University of Toronto's main and Scarborough campuses.

Thus ChalkZone took such alternate universe stories as Through the Looking Glass and The Wizard of Oz one step further.

" Through their correspondence they increased the awareness of women's issues, were supportive, and influenced the course of events to further America's cause.

Through cultural assimilation the Simurgh was introduced to the Arabic-speaking world, where the concept was conflated with other Arabic mythical birds such as the Ghoghnus, a bird having some mythical relation with the date palm, and further developed as the Rukh ( the origin of the English word " Roc ").

Through further mergers and acquisitions, Lord ultimately became president of the British Motor Corporation.

Through Maderna, he became acquainted with Hermann Scherchen — then Maderna's conducting teacher — who gave Nono further tutelage and was an early mentor and advocate of his music.

Through his father, he was then employed for a further three months in Hong Kong by Jardine Matheson as a ' ship jumper ', an administrative post.

Through various ticket consolidators, charter airlines, and innovators in lower frills flying, such as Channel Airways, and Court Line, the traveling public had been conditioned to want to travel to new and increasingly further away and exotic locations on vacation, rather than short-haul trips to nearby beach resorts.