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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

129 integ below is a partial table of integr als of

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Unformatted text preview: ulus. Gottfried Wilhelm Leibniz (1646-1716) in his manuscript dated 29 October 1673, was the first to use the integral sign ∫, the elongated or stretched S continuous as we know and use it today, to denote continuous summation. Leibniz called contin the expression ∫ f (x) dx the INTEGRAL from the Latin integralis o r whole. i ntegralis However, both the concepts of defining the r elationship between Differential and Integral Calculus: DERIVATIVE & ANTIDERIVATIVE and TANGENT and “ ar ea under the cur v e ”, are useful but limited. The correct concept that corresponds to the relationship is INSTANTANEOUS RATE OF CHANGE and CHANGE. 126 Over view Ov er vie w differentia We may define a WELL-BEHAVED function F(x) and on differentiation let : dif entiation d F(x) f(x) = dx Hence f(x) is the DERIVATIVE of F(x). If we write f(x)dx = dF(x) then we call f(x) the DIFFERENTIAL COEFFICIENT. inverse opera differentia We may now define the inver se oper ation of dif f er entiation as finding the in dif entiation ANTIDERIVATIVE . So : F(x) = ANTIDERIVATIVE { f(x)} differentia Just like with differentiation we may construct a table of ANTIDERIVATIVES. The dif entiation calculation becomes almost mechanical. This is purely from the calculation point calculation of view. From the Analysis point of view we may infer that : if f(x) is the expression of the Analysis INSTANTANEOUS RATE OF CHANGE of F(x), then F(x) must be the expression of CHANGE. From the Geometric point of view we have an even more interesting relationship. It Geometric seems almost magical. area under f(x) CHANGE in F(x) = = F(b) − F(a) over interval [a, b] over interval [a, b] Integral Calculus developed out of the need for a general method to find areas, volumes and centres of gravity. Computing the area under f(x) involves a continuous summation . It is from this process of continuous summation that continuous we have the operation expressed as integration derived from the Latin integralis integration integralis and the Mathematical notation or symbol ∫ to def...
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