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Gottfried Wilhelm Leibniz (16461716) 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 WELLBEHAVED 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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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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