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in 1635 and 1647, in which by ingenious use of certain cases of the
Binomial Theorem he obtained the area under
y
x
and
y
x
, and
suggested a formula for the area under
y
x
for positiveintegral
n
.
Proving this result and extending it to all rational
n
1 (as in qns
3
—
5) is commonly credited to Fermat who claimed the result in general
form, about 1643, though Torricelli also obtained the result about this
time. The connection between the area under a rectangular hyperbola
and logarithms was identified by Gregory of St Vincent (1647) and
A. A. deSarasa (1649).
In a topic written by Isaac Barrow, Newton's predecessor as
Lucasian professor at Cambridge, and published in 1669, while
discussing continuous and monotonic distance
—
timeand vlocity
—
time
graphs, Barrow described the Fundamental Theorem of Calculus. The
description is geometrical (and therefore, from a modern viewpoint,
hard to recognise) and the proof uses infinitesimals. Newton helped
Barrow to write the topic, and there was a copy of it in Leibniz'
personal library. From his papers, we know that Newton came to
understand the Fundamental Theorem in the year 1665
—
6. He
considered two variables
x
and
y
(which he called fluents) changing
with time. Their rates of change he denoted by
x
and
y
(which hecalled
fluxions) and heobtained
A
/
x
y
, where
A
is the area under the graph
of
y
against
x
. Newton's discovery of the inverse relationship between
integration and differentiation, or as he would have put it, between the
method of quadratures and the method of fluxions, transformed the
study of areas and generated a range of powerful applications. Newton's
readiness to differentiate and integrate power series term by term
(without tests of convergence) brought a further host of significant
results relating to the binomial series and trigonometric functions.
Leibniz obtained Newton's fundamental theorem independently in 1675.
In his
Principia
(1687), Newton included a proof that any monotonic
function was integrable. His illustrations show that he was considering
only differentiable monotonic functions, but all the functions hewas
concerned with were piecewise monotonic, so his proof dealt
satisfactorily with his field of concern. Quite trivially, his proof may be
extended to show that discontinuous monotonic functions are
integrable, but this was not done until the integrability of such
functions was considered in the second half of the nineteenth century.
The wide application of this result and the fact that, in the century
following Newton and Leibniz, functions were considered to be
infinitely differentiable meant that, during the eighteenth century, the
study of integration consisted of the search for anti-derivatives. It was
Cauchy (1823) who shifted his attention from the indefinite to the
definite integral for the purpose of defining the integral of a continuous
