Calculus: Analytic Geometry and Calculus, with Vectors

7.2 Lengths and integrals 419

This is, in fact, a correct and very satisfying formula which treats all
coordinates alike and enables us to calculate lengths of curves that wind
around through E3. In case C lies in the xy plane, everything is the same
except that z(t) = z'(t) = 0 and the formula is thereby simplified. In
applications, it often happens that x(t) = t and the formula is written
in one or anotherr of the forms

(7.241) SCI =

J

b 1 + [y'(t)l2 + [ (t)]2 dt

a

_ f b

(^1) [[y'(x)12 + [z'(x)]2 d.r
fb
a
= 2 2
1 -I- ()
+() dx
and, for plane curves, we have z = 0. It is not easy to be sure that the
"cheerful derivation" of (7.24) is slovenly. A person can claim that he
will not use the definition of length given in Section 7.1, that he is interested
only in curves C for which x'(t), y'(t), and z'(t) are continuous, and that
all of the work that he has done merely motivates his definition whereby
JCJ is defined by (7.24). His position is tenable and, so long as he stays
in his own garden, it is even reasonable. In fact, many quantities in
pure and applied mathematics are defined by integrals, and sometimes
the definitions are so well motivated that readers (and even writers) fail
to recognize the fundamental fact that it is utterly impossible to prove
correctness of a formula for something that has not been defined.
We start again with the formula

()2

(7.25) ICI = lim + + Az Atk,

(Eik_) k2

()2

which is (7.232) brought up where we can see it, but this time our situa-

tion is different. We suppose that JCJ is defined by (7.25), and we want

to prove that the right side is an integral. For present purposes, the work

of the preceding paragraph would be slovenly because nothing was proved.

We could try to fix everything by trying to prove that there is a tkti* such

that, with the notation of the preceding paragraph, the kth term of

(7.232) is exactly equal to f (tR*,) Atk. We do not try this, however, because

of the fear that the process would involve only uninformative hard work.

Instead, we start by applying the mean-value theorem (Theorem 5.52)

to (7.25). Since x' is continuous, there must be a tk for which tk-1 <

tk < tk and

(7.251) L\xk=X(tk) - x(tk-1) = x (t*).

Atk tk - tk-1 k

Similar formulas involving y and z end with y'(tk *) and z'(tk***), there