This preview shows page 1. Sign up to view the full content.
Unformatted text preview: P = { a = t < t 1 < ··· < t n = b } . ±or a small mesh size, the arclength of C between successive points −→ p ( t j ) is well approximated by △ s j = b −→ p ( t j − 1 ) − −→ p ( t j ) b and we can form lower and upper sums L ( P ,f ) = n s j =1 inf t ∈ I j f ( −→ p ( t )) △ s j U ( P ,f ) = n s j =1 sup t ∈ I j f ( −→ p ( t )) △ s j . As in the usual theory of the Riemann integral, we have for any partition P that L ( P ,f ) ≤ U ( P ,f ); it is less clear that reFning a partition lowers U ( P ,f ) (although it clearly does increase L ( P ,f )), since the quantity ℓ ( P , −→ p ) increases under reFnement. However, if the arc is rectiFable, we can modify the upper sum by using s ( −→ p ( I j )) in place of △ s j ; denoting this by U ∗ ( P ,f ) = n s j =1 sup t ∈ I j f ( −→ p ( t )) s ( −→ p ( I j ))...
View
Full
Document
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Integrals

Click to edit the document details