Calculus: Analytic Geometry and Calculus, with Vectors

7.2 Lengths and integrals 421

all have absolute values less than e' when I u - oil < S. When JP1 < S,

the right member of (7.255) then has the form

(7.263) Sk = VA k ± h - flk,

where the quantities under the radicals are nonnegative and 0 < h _<_ 36'.

It follows that 141 < 3e' and hence

n n

(7.264) Sk Atk I < NI-3-W At, = 3E (b - a) < e

k=1 k=1

provided e' is so chosen that the last inequality holds. This proves
(7.261) and hence (7.26).
In order to introduce coordinates on curves, we suppose that x, y, and
z are functions having continuous derivatives over an open interval
a < t < b and that C is the curve traversed
by the point P(t) having coordinates (x(t), s=7 s=8
y(t), z(t)) as t increases. To avoid difficul-
ties, we suppose that the curve is simple;
this means that P(ti) P(t2) when tl 0 t2
so that P(t) cannot be in the same place at

s

S=6

two different times. Let to be fixed such

that a < to < b. Then, as in Figure 7.27,

we can assign coordinates to points on C by

letting s be the length of the curve running

from P(to) to P(t) when to < t < b and

Figure 7.27

letting -s be the length of the curve running from P(t) to P(to) when

a < t 5 to. Then, whenever a < t < b, the coordinate s(t) of P at time t is

(7.28) s(t) = fot [x (r)]2 + [y'(r)}2 + [z (r)}" dr.

In this formula r (tau, the Greek t) is used as a dummy variable of integra-

tion. Since our hypothesis implies that the integrand is continuous,

Theorem 4.35 enables us to differentiate with respect to t to obtain the

formula

(7.281) s'(t) = [x'(t)]2_[y1(t)j1+[z'(t)]2,

which is often written in the form

(7.282)

at=

Cax\ 2

+dt )2+

(dd

2

This is related to the formulas we get when we set

(7.283) r = x(t)i + y(t)j + z(t)k