On This Day
1675 – Gottfried Leibniz demonstrates integral calculus for the first time to find the area under the graph of y = ƒ(x).
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral of f from a to b can be interpreted informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. It is denoted
∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} \int _{a}^{b}f(x)\,dx.
The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, called an indefinite integral, a function F whose derivative is the given function f. In this case, it is written:
F ( x ) = ∫ f ( x ) d x . {\displaystyle F(x)=\int f(x)\,dx.} F(x)=\int f(x)\,dx.
The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval [a, b], then once an antiderivative F of f is known, the definite integral of f over that interval is given by
∫ a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}\,f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)\,.} {\displaystyle \int _{a}^{b}\,f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)\,.}
The principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous mathematical definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the 19th century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalized. A line integral is defined for functions of two or more variables, and the interval of integration [a, b] is replaced by a curve connecting the two endpoints. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.
1936 – In California, the San Francisco–Oakland Bay Bridge opens to traffic.[9]
The San Francisco–Oakland Bay Bridge, known locally as the Bay Bridge, is a complex of bridges spanning San Francisco Bay in California. As part of Interstate 80 and the direct road between San Francisco and Oakland, it carries about 260,000 vehicles a day on its two decks.[3][4] It has one of the longest spans in the United States.
The toll bridge was conceived as early as the California Gold Rush days, but construction did not begin until 1933. Designed by Charles H. Purcell,[6][7] and built by American Bridge Company, it opened on Thursday, November 12, 1936, six months before the Golden Gate Bridge. It originally carried automobile traffic on its upper deck, with trucks, cars, buses and commuter trains on the lower, but after the Key System abandoned rail service, the lower deck was converted to all-road traffic as well. In 1986, the bridge was unofficially dedicated to James Rolph.[8]
The bridge has two sections of roughly equal length; the older western section, officially known as the Willie L. Brown Jr. Bridge (after former San Francisco Mayor and California State Assembly Speaker Willie L. Brown Jr.), connects downtown San Francisco to Yerba Buena Island, and the newer unnamed eastern section connects the island to Oakland. The western section is a double suspension bridge with two decks, westbound traffic being carried on the upper deck while eastbound is carried on the lower one. The largest span of the original eastern section was a cantilever bridge. During the 1989 Loma Prieta earthquake, a portion of the eastern section’s upper deck collapsed onto the lower deck and the bridge was closed for a month. Reconstruction of the eastern section of the bridge as a causeway connected to a self-anchored suspension bridge began in 2002; the new eastern section opened September 2, 2013, at a reported cost of over $6.5 billion, a 2,500% cost overrun from the original estimate of $250 million (for a seismic retrofit of the existing span).[9][10] Unlike the western section and the original eastern section of the bridge, the new eastern section is a single deck with the eastbound and westbound lanes on each side making it the world’s widest bridge, according to Guinness World Records,[11] as of 2014. Demolition of the old east span was completed on September 8, 2018.[12]
Born On This Day
1866 – Martha Annie Whiteley, English chemist and mathematician (d. 1956)[5]
Martha Annie Whiteley, OBE FCS (11 November 1866 – 24 May 1956)[1] was an English chemist and mathematician. She was instrumental in advocating for women’s entry into the Chemical Society, and was best known for her dedication to advancing women’s equality in the field of chemistry.[2] She is identified as one of the Royal Society of Chemistry’s 175 Faces of Chemistry.[3]
1905 – Louise Thaden, American pilot (d. 1979)[18]
Iris Louise McPhetridge Thaden (November 12, 1905 – November 9, 1979) was an American aviation pioneer, holder of numerous aviation records, and the first woman to win the Bendix trophy, alongside Blanche Noyes.[1] She was inducted into the Arkansas Aviation Historical Society’s Hall of Fame in 1980.[2]
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FYI
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NSFW
https://youtu.be/5PrLGhJnO7I
Recipes
By breadandbananas: Cong You Bing Croissants
By Regan_Jane: Truffles Coated in Freeze Dried Raspberry Powder
RecipeTinEats
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wikimedia
- Wikimedia Commons picture of the day for October 4 October 4, 2024Picture of the day Crucifix and shadow on the wall, Abbey of Saint Peter (Assisi). Today is the feast of Francis of Assisi, Patron Saint of Ecologists.
- Wikimedia Commons picture of the day for October 3 October 3, 2024Picture of the day Philippsruhe castle in Hanau-Kesselstadt, seen from the west from the park.
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