144 Chapter 5Integration
Following Leibniz, this limit is written as
(5.33)
and is called the definite integral of the functionf(x)fromx 1 = 1 atox 1 = 1 b.
This taking of the limit of the sum of small quantities is the characteristic feature
of the integral calculus, just as taking the limit of the ratio of two small quantities is
characteristic of the differential calculus. The essential discovery, made independently
by Leibniz and by Newton, was that ifF(x)is a function whose derivative isf(x) 1 = 1 F′(x)
then
(5.34)
This synthesis of the integral and differential calculus is called the fundamental
theorem of the calculus. The definite integral as defined in this section is known as
the Riemann integral.
3In calculating the area enclosed by a curve it is neither essential nor always
convenient that the area be divided into linear strips in the way described above. We
illustrate this point in Example 5.11 with two different ways of calculating the area of
a circle.
EXAMPLE 5.11The area of a circle
The equation of the circle of radius aand centre at the origin of coordinates is
x
21 + 1 y
21 = 1 a
2.
Method 1.Let Abe the area of that quarter of the circle that lies in the first quadrant,
in which both xand yare positive (Figure 5.12). Then
Divide the area into nvertical strips as described in
the derivation of equation (5.33) above. An approxi-
mate value of the area of the strip between xand
x 1 + 1 ∆xis and, by equation
(5.33), the total area is
Aaxdx
a=−Z
022∆∆Ayx a x≈=−∆x
22yax=− ≤≤.xa
22for 0
ZZ
ababf x dx() = F x dx F b F a′() =−() ()
Z
abrnr rfxdx fx x
n
( ) = lim ( )
→
=∑
∞
1∆
x
y
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0 x x+∆ x a
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Figure 5.12
3Georg Friedrich Bernhard Riemann (1826–1866), professor of mathematics at Göttingen, made contributions
to the theory of numbers, functions of a complex variable, and differential equations. In his 1854 lecture Über die
hypothesen welche der geometrie zu grunde liegen(On the hypotheses that lie at the foundations of geometry)
Riemann developed a system of non-Euclidean geometry and initiated the study of curved metric spaces that
ultimately formed the basis for the mathematics of the general theory of relativity.