Calculus: Analytic Geometry and Calculus, with Vectors

422 Curves, lengths, and curvatures

and calculate the velocity v and the speed (vI from the formulas

(7.284) v = x'(t)i + y'(t)j + z'(t)k

and

(7.285) IvI = [x (t)]'` + [y'(t)]2 -]- [z'(t)]2.

Since the right members of (7.282) and (7.285) are the same, this shows

that, in appropriate circumstances, the speed determined so quickly by

taking the absolute value of the velocity is the same as the "speed of the

particle in its path" determined by use of coordinates on the path.

Problems 7.29

1 Find the length of the part of the helix

x=acost, y=asint, z=kt

traversed by a particle at P(x,y,z) as t increases from 0 to 27r.

2 Using the first standard equation in

2 2

a2-}-b2=1, x=acos6, y=bsin6,

where 0 < b < a, set up an integral for the length of the part of the ellipse lying

in the first quadrant. Then use the parametric equations to set up another

integral for the same length

Ins.: f a

aa2

-kx dx, af 'r" 1 - k2 cos2 8 do,

where k2 = (a2 - b2)/a2. These are elliptic integrals, and others more or less

like them are also called elliptic integrals. There is no easy way to obtain their

exact numerical values.

3 Set up an integral for the length L of the curve traced by the point P

having coordinates

x=a6-bsin0, y=a-bcos0

as 0 increases from 0 to 2n, and show that the result can be put in the form

L = (a + b) f

02,

I1 (a+bb)2 cos2^9 d8.

When b 0 a, this is an elliptic integral. When b = a, the graph is, as Problem

16 of Section 6.5 shows, an ordinary cycloid. Sketch a figure and explain a

simple geometric argument which shows that, when b = a and the approximations

x = 3 and 72 = 10 are used, 52 a < L < 10a. Finally, show that L = 8a.

4 Find the length of the part of the curve having the equations

+ 2

x=2ta, y=t^2 ' z=

2t

which lies between the points (0,-I,1) and (2,-1,$), fins.: L =Z[1931- 1].