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History of integration
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
v Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a functionƒ of a realvariablex and an interval [a, b] of the real line, the definite integral
is defined informally to be the net signed area of the region in the xy-plane bounded by the graphof ƒ, the x-axis, and the vertical lines x = a and x = b.
The term integral may also refer to the notion ofantiderivative, a function F whose derivative is the given function ƒ. In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.
The principles of integration were formulated independently by Isaac Newton and GottfriedLeibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by
Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth 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 generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curveconnecting two points on the plane or in the space. In asurface integral, the curve is replaced by a piece of a surfacein the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integral first arose from the needs ofphysics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known asLebesgue integration, developed by Henri Lebesgue.
History ofquadratic equation and Quadratic Function
On clay tablets dated between 1800 BC and 1600 BC, the ancient Babylonians left the earliest evidence of the discovery of quadratic equations, and also gave early methods for solving them.
Indian mathematician Baudhayana who wrote a Sulba Sutra in ancient India circa 8th century BC first used quadratic equations of the form
ax²=c and ax²+bx=c and also gave methods for solving them.
Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula.
Euclid, a Greek mathematician, produced a more abstract geometrical method around 300 BC.
The first mathematician to have found negative solutions with the general algebraic formula was Brahmagupta (India, 7th century).
Muḥammad ibn Mūsā al-Ḵwārizmī (Persia, 9th century) developed a set of formulae that worked for positive solutions.
Bhaskara II (1114-1185), an Indian mathematician-astronomer, solved quadratic equations with more than one unknown and is considered the originator of the equation.
Shridhara (India, 9th century) was one of the first mathematicians to give general rule for solving a quadratic equation.
Applications of Quadratic Function
The sum of two integers is 30. What is a maximum product of these two numbers?
Using L and S as the larger and smaller numbers respectively, we get
The maximum value of the product can be found after writing the product as either a function of L or of S. Writing P as a function of L we get
. The graph of this function in an L, P coordinate system, where L stands for the larger number and P for the product is:
Using the vertex formula, with a = -1 and b = 30 we get the coordinates of the vertex as (15,
225). So the maximum value of the product is 225 for L = 15, and this makes S = 15 as well.
ha...sampai sini je oren buat...nak keje2.... nanti siri ke 2 nanti oren sambung lagi..hmmm....mintak tolong cik nab buatkan innit laah...trimasss~~