Calculus: Analytic Geometry and Calculus, with Vectors

252 Integrals

are defined by these formulas whenever the integrals on the right exist as

Riemann-Cauchy integrals. Perhaps attention should be called to the

fact that some elementary books reserve the term "definite integral"

for application to an integral of a particular brand (which is sometimes

0

Figure 4.631

the Riemann brand and is sometimes not care-

fully delineated) and apply the term "improper

integral" to each integral of another kind.

It is impossible to have a tranquil scientific

career without thorough understanding of

matters relating to

(4.63)

• 1
  1 2 dx.
  lx

The graph of the integrand is shown in Figure

s 4.63 1. The integral cannot exist as a Riemann

integral because the integrand x-1 is undefined

when x = 0. Even if we set f (O) = 0 and

f (x) = x2 when x 0, the integral f

1

1 f (x) dx will still fail to exist as

a Riemann integral because f is not bounded over the interval

-1 < x < 1. According to (4.624), the formula

(4.632) J11 x2 dx= Jof x2 dx J x2dx

will be valid when the integrals are Riemann-Cauchy integrals provided

the two integrals on the right side exist. The calculation

(4.633) J

of

x2 dx= hl-lo+ f

__1h

x2dx = lo+ 1i - 1

= 00

shows that the first integral on the right does not exist, and the calculation

(4.634) f 1

x2

dx = lim

J

1 1 dx= lim 11- 1 I= ao

o h-+0+ h x2 h-o+ h

shows that the second does not exist. Hence, the integrals in (4.632) do

not even exist as Riemann-Cauchy integrals. The calculations do, how-

ever, enable us to convey information by writing

(4.635)

f-1

2

dx dx +

fil

2 dx ao.

x lx x

Persons do not lead these tranquil scientific lives when they realize that

(4.636)

d -11= 1

dx x x2