This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 20E Homework #4 27 January 2005 Section 4.2, Problem 2: Find the arc length of the curve ( 1, 3 t 2 , t 3 ) over the interval t 1. Solution. The arc length is Z 1 k ( 0, 6 t , 3 t 2 ) k dt = Z 1 p 36 t 2 + 9 t 4 dt = Z 1 3 t p 4 + t 2 dt = ( 4 + t 2 ) 3/2 1 = 5 5 8. Section 4.2, Problem 10: The arc length function s ( t ) for a given path c ( t ) , defined by s ( t ) = R t a k c ( ) k d , represents the distance a particle traversing the trajectory of c will have travelled by time t if it starts out at a ; that is, it gives the length of c between c ( a ) and c ( t ) . Find the arc length functions for the curves ( t ) = ( cosh t , sinh t , t ) and ( t ) = ( cos t , sin t , t ) , with a = 0. Solution. For ( t ) , we get s ( t ) = Z t q sinh 2 + cosh 2 + 1 d = Z t r 2 e 2 + 2 e 2 4 + 1 d = 2 Z t e + e 2 d = 2 sinh t . For ( t ) we get s ( t ) = Z t q ( sin ) 2 + cos 2 + 1 2 d = Z t 2 d = 2 t . Section 4.2, Problem 11: Let c ( t ) be a given path, a t b . Let s = ( t ) be a new variable, where is a strictly increasing C 1 function given on [ a , b ] . For each s in [ ( a ) , ( b )] there is a unique t with ( t ) = s . Define a function d : [ (...
View
Full
Document
 Fall '07
 Enright
 Arc Length, Vector Calculus

Click to edit the document details