Calculus: Analytic Geometry and Calculus, with Vectors

418 Curves, lengths, and curvatures

could be supposed that our main interest is in lengths of curves and that

we should get our result as quickly as possible, but this is not true.

Integrals are very important things, and we proceed slowly to learn as

much as we can about them.

Our first step is to make a partition P of the interval a _<_t <_ b having

partition points to, to, ,'. , tand to sketch schematic figures showing

the partition and points Pk-1 = P(tk-1) and Pk = P(tk) on C. We then

tell ourselves that if API, the norm of the partition, is small then IPk_1pk'

should be a good approximation to the length of the part of C between

PA_i and PA, so 2;';PL_1PkI should be a good approximation to ICI and hence

it should be true that

(7.23) ICI = lim ZIPk-1PkI

We have employed the fundamental ideas about length given in Section

7.1. The next step is to put the right member into a more useful form.

While introduction of the abbreviations may be unwise when acres of

paper and blackboard space are available, we simplify our formulas by

setting

Otk = tk - tk-1, Oxk = X(tk) - x(tk-1), AYk = Y(tk) - Y(tk-1),

A2k = z(tk) - Z(tk-1).

Then (7.23) gives

(7.231) ICI = Jim Axk Ay, + AZk

We can make this look much more like a limit of Riemann sums by

introducing factors i tk in numerators and denominators to obtain

(7.232) ICI = llm L]xk2+ .yk/2 +

CAZk/2ltk.

4 l Atk Otk

The possibility of making further progress is provided by the hypothesis

that x, y, and z have continuous derivatives.

We should be realistic and realize that there are different ways to

proceed. We can cheerfully adopt the view that, however we choose

t* in the kth subinterval of our partition, the kth term in (7.232) should

be closely approximated by f(t,*) Otk, where

(7.233) f(t) = [x'(t)]2 + [Y (t)]2 + [z'(t)]2.

Thus we can expect that

(7.234) ICI = lim lf(tk) Atk.

Since our hypothesis guarantees that f is continuous and hence integrable,

the Riemann sums do have a limit and we are led to the formula

(7.24) ICI =

ffb-,1[X1(t)j2

+ [Y'(t)l2 + [z'(t)]2 dt.