Page 99 - Schaum's Outline of Theory and Problems of Advanced Calculus

Integrals

INTRODUCTION OF THE DEFINITE INTEGRAL
The geometric problems that motivated the development of the integral calculus (determination of
lengths, areas, and volumes) arose in the ancient civilizations of Northern Africa. Where solutions were
found, they related to concrete problems such as the measurement of a quantity of grain. Greek
philosophers took a more abstract approach. In fact, Eudoxus (around 400 B.C.) and Archimedes
(250 B.C.) formulated ideas of integration as we know it today.
Integral calculus developed independently, and without an obvious connection to differential
calculus. The calculus became a ‘‘whole’’ in the last part of the seventeenth century when Isaac Barrow,
Isaac Newton, and Gottfried Wilhelm Leibniz (with help from others) discovered that the integral of a
function could be found by asking what was differentiated to obtain that function.
The following introduction of integration is the usual one. It displays the concept geometrically and
then defines the integral in the nineteenth-century language of limits. This form of definition establishes
the basis for a wide variety of applications.
Consider the area of the region bound by y ¼ f ðxÞ, the x-axis, and the joining vertical segments
(ordinates) x ¼ a and x ¼ b. (See Fig. 5-1.)

y

y = f (x)

x
ξ n b
a ξ 1 x 1 ξ 2 x 2 ξ 3 x 3 x n _ 2 x n _ 1
ξ n _ 1
Fig. 5-1
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