5.2 The indefinite integral 127
It is readily verified that this distance is equal to the area, shaded in the figure, bounded
by the linev(t)and the t-axis betweent
Aandt
B. We shall see that this last result is valid
for any velocity functionv(t). We shall also see that the solution of a physical problem
is often equivalent to finding the area enclosed by an appropriate curve.
This example demonstrates the two central problems of seventeenth-century
European mathematics; the ‘problem of tangents’ and the ‘problem of quadrature’.
The first of these, to find the tangent lines to an arbitrary curve, led to the invention of
the differential calculus, the subject of Chapter 4. The second, to find the area enclosed
by a given curve, led to the invention of the integral calculus.
2The demonstration by
Leibniz and by Newton that differentiation and integration are essentially inverse
operations is one of the landmarks of the history of mathematics.
The concept of integration as the inverse operation to differentiation leads to the
definition of the indefinite integral. The concept of the integral as an area leads to the
definition of the definite integral.
5.2 The indefinite integral
Lety 1 = 1 F(x)be a function of xwhose derivative is. The indefinite integral
of the derivative is defined by
Fx′ =
dy
dx
()
.......................................................................................................................................................................................................................................
...
..
...
...
..
...
..........................................v
Av
Bt
At
Bv
t
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................................................................................................................................................................................................................................
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................
..
Figure 5.2
2Integration has its origins in the Greek ‘method of exhaustion’ for finding areas and volumes. Archimedes
in his Quadrature of the parabolaattributed the method to Eudoxus of Cnidus (c. 408–355 BC). In the Method,
discovered in Constantinople in 1906 after being ‘lost’ for over a thousand years, Archimedes describes how a
plane area can be regarded as a sum of line segments. In 1586, Stevin described how the centroid of a triangle can
be obtained by considering the area as made up of a large number of parallelograms. Johann Kepler (1571–1630),
best known for his Astronomia novaof 1609, computed areas and volumes by considering them to be composed
of infinitely many infinitesimal elements. His work on volumes appeared in 1615 in Nova stereometria doliorum
vinariorum(New solid geometry of wine barrels). Galileo Galilei (1564–1642) made use of the infinitely small in
his work on dynamics. Bonaventura Cavalieri (1598–1647), a follower of Galileo, described in his influential
Geometria indivisibilibus continuorum, 1635, how an area can be thought of as made up of lines or ‘indivisibles’
and a volume of areas, and developed a geometric method for finding the integral ofx
nfor positive integers n. At
about the same time, Fermat solved the same problem for positive and negative integers (exceptn 1 = 1 − 1 ) and for
fractions by dividing his areas into suitable rectangular strips. The case ofn 1 = 1 − 1 was treated by Gregoire de Saint
Vincent (1584–1667). Roberval integrated the sine function in 1635, and Torricelli the log function in 1646. Other
contributors include Pascal, whose Traité des sinus du quart de cercleof 1658 Leibniz said inspired his discovery of
the fundamental theorem, John Wallis (1616–1703), whose work on infinite processes influenced Newton and to
whom we owe the symbol ∞, the Scot James Gregory (1638–1675) whose work on infinite series and the calculus
anticipated that of Newton, and Barrow, whose lectures Newton attended and a copy of whose Lectioneswas
bought by Leibniz when on a visit to London in 1673. The final step in the synthesis of the differential and integral
calculus was taken by Newton (Footnote 3, Chapter 4) and by Leibniz, who published the first account of his
integral calculus, Analysi indivisibilium atque infinitorum(Analysis of indivisibles and infinities) in 1686.