13.3: Arc Length and Speed

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

89

Pages:

2
Author:

Alexis H.

13.3 Arc Length and speed Recall that arc length is defined as the limit of the lengths of polygonal approximations. Let C be the curve parametrized by a·t·b r (t) = hx (t) ; y (t) ; z (t)i; We assume that the parametrization directly traverses C; that is, the path traces C from one end to the other without changing direction along the way Theorem 1: Arc length Let r(t) directly traverse C for a < t < b. Assume r’(t) exists and is continuous. Then the arc length s of C is equal to s= Z b 0 kr (t)k dt = a Z b a q 2 2 2 x0 (t) + y 0 (t) + z 0 (t) dt Example: Find the arc length s of the helix given by r(t) = for 0 < t < 2pi r0 (t) = h¡3 sin 3t; 3 cos 3t; 3i s= ¡ ¢ 2 kr0 (t)k = 9 sin2 3t + 9 cos2 3t + 9 = 9 sin2 3t + cos2 3t + 9 = 18 Z 2¼ 0 kr (t)k dt = 0 Z 2¼ p p 18dt = 6 2¼ 0 Speed, by definition, is the rate of change of distance traveled with respect to time t. To calculate the speed, we define the arc length function: s (t) = Z t kr0 (u)k du a This relationship provides us with the second of the two important features of the of the velocity vector r’ (t): it points in the direction of motion, and its magnitude is the speed V (t) = r0 (t) ; velocity vector v (t) = kV (t)k speed Arc Length Parametrization Given a parametrization r(t) we obtain a new parametrization by making a substitution t = g (u) — that is, by replacing r(t) with r1 (t) = r (g(u)) A way of parametrizing a curve that is useful in studying properties of the curve is to choose a starting point and travel along the curve at unit speed. A parametrization of this type is called an are length parametrization with the parameter often denoted by s There are three important properties of an arc length parametrization: The parameter s corresponds to the arc length of the curve that is traced from the starting point (and thus the name arc length parametrization) Every velocity vector r’(s) has length equal to 1 The arc length of the curve that is traced over any interval [a, b] is equal to b - a, the length of the interval: Distance traveled over [a, b] = Z b 0 kr (s)k ds = a Z b 1dt = b ¡ a a