Calculus: Analytic Geometry and Calculus, with Vectors

4.2 Riemann sums and integrals 217

So far, the integral in (4.271) has been defined only when x > a.

now completethe definition by setting

We

(4.281) f as f (t) dt= 0,

fax

f (t) dt = -

f a

f (t) dt,

the second formula being valid when x < a and f is integrable over the
interval from x to a.

Problems 4.29

1 Practice the art of telling how the number

LX f(i)dt

is defined. Be prepared to give the full details, including Riemann sums, at
any time.
2 Tell whether you think it wise to abbreviate the statement "To each
positive number e there corresponds a positive number S such that

[nn

Ljlf(tk) At, - far' f(t) di I < e

whenever the sum is a Riemann sum formed for a partition P of the interval

a < t _< x having norm less than S" to the statement

lim f(tk) Itk = f X f (t) dt.

PI-0 k=1

a

Remark: If you do not have an opinion, think about the matter and get one.

3 For better or for worse, the "formula"

limIf(tk)L1tk =

faXf(t)

dt

is considered to be an assertion. Tell precisely what it means.

4 Tell whether you would like to learn and use a completely new notation by

which the "formula"

approxI f (tk) L1 tk = f

x

f (t) dt

eJPI<5 k=1 a

is used to abbreviate the statement that to each positive number a there corre-

sponds a positive number S such that

I f(tk) Otk -fax f(t) dt I < e

k=1

whenever P is a partition of the interval a 5 t<-- x for which JPJ < S. Remark:

If you do not have an opinion, think about the matter and get one.

5 We often hear about the great scientific progress of our modern era,

and we should think about an example. One of the great strides forward is made