Calculus: Analytic Geometry and Calculus, with Vectors

410 Curves, lengths, and curvatures

as we think of them asbeing traversed once in the positive (counter-

clockwise) or negative (clockwise)direction.

Supposing that C is a given curvefor which the interval appearing in

Definition 7.14 is a closed interval a < t < b, we proceed to define (when

it exists) a number ICI whichis called the length of C. Let P be a parti-

tion of the interval by partitionpoints to, t1j , tofor which

(7.15) a=to<tl<t2< <tn_1<tn=b.

For each k = 0, 1, 2, , n,let Pk be the point on C for which t = tk.

Figure 7.151

The number SP defined by

k-1 to P2 and so on to P. is said to be inscribed in C.

Pk polygon (or broken line) running from Po to P1

Figure 7.151 is a schematic figure that may be

helpful, but it is much too simple to show how

much care is needed to guarantee that the points

are not scrambled in an inappropriate way. The

n

SP = E IPk_IPkI

k=1

is the sum of the lengths of the sides of the inscribed polygon. Let S

be the set of numbers Sp obtained by making partitions P of the interval

a <_ t <_ b. In case there is no number M such that Sp < M whenever

SP is in S, it is said that C does not have length or that C does not have

finite length or that C has infinite length or that Cl I= o o. In case there

is a number M such that SP < M whenever SP is in S, then Theorem 5.46

guarantees existence of a least number ICI such that Sp < ICI whenever

SP is in S. This number ICI is then, by definition, the length f of C.

For future reference we state without proof the following theorem which

involves notation we have been using.

Theorem 7.16 The curve C has finite length ICI if and only if

n

ICI = Jim I, IPk_1PkI.

IPI-'O k-1

We now use the definition of length to prove the following theorem

which gives, as a corollary, the desired result involving Figure 7.11.

Theorem 7.17 Let f be continuous and monotone increasing (or mono-

tone decreasing) over a < x <_ b. Let C be the graph of y = f(x) so oriented

t In old books particularly, a curve C having finite length is sometimes said to be recti-

fiable; the ancient idea is that if C is a string we could pull on the ends and straighten it out

to get a straight string of finite length. We can all know enough about logic to know that

if C is not a string, then little credence is placed upon consequences of the false assumption

that C is a string.