Calculus: Analytic Geometry and Calculus, with Vectors

4.6 Riemann-Cauchy integrals and work 251

exists as a Riemann integral whenever h? a and that this integral, as a
function of h, has a limit as h becomes infinite. In such cases, this limit

is the Riemann-Cauchy integral off over the semi-infinite interval x > a
and is denoted by the symbol in the left member of (4.61). In each other
case, we say that the integral in the left member of (4.61) does not exist
as a Riemann-Cauchy integral. For example, when r > 0,

(4.611) xdx=h^1 1]^1

r x 2dx=lim-l]=limrr -h

f
L J
We can bravely start to calculate a Riemann-Cauchy integral by
tentatively writing

h h

(4.612)

o°°

cos x dx = limf o cos x dx = lim [sin x]o = lim sin h

h-+ m h-. m h- o

with the understanding that we will get an answer if the last limit exists.

The last limit does not exist, however, so the integral does not exist.

Riemann-Cauchy integrals of another type are defined by the formula

(4.62) f oa f (x) dx= lira f ha f (x) dx

h-+0+

when a > 0 and the integrals and limit exist. Consider the example for

which f(x) = x-34 when x > 0, while f(x) is either undefined or is defined

in some other way when x < 0. Then f is not bounded over the interval

0 < x 5 1 and hence L' f(x) dx cannot exist as a Riemann integral.

However,

(4.621) f 1 x-34 dx= lim

1

0 x 3 dx

h-o+ h

= lim 2x34 lim [ 2- 21/h] = 2,

h-O+

so the first integral exists as a Riemann-Cauchy integral. Riemann-

Cauchy integrals of still other types are defined by the formulas

(4.622) f

0

af(x) dx = lim f -h f(x) dx

h-0+ -a

f a.

f (x) A = lim f ha f (x) dx

h-+ - m

when the integrals and limits exist. Finally, the Riemann-Cauchy

integrals in the left members of the formulas

(4.623) f. f(x) dx = f a. f(x) dx +f f(x) dx

(4.624) f ab f (x) dx =f a` f (x) dx + f eb f(x) dx