Tutorial08-sol(1)

# Here 0 is a parameter and the curve c is oriented

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: C where C is the half of Viviani’s curve 2 + 2 + 2 = 2 , 2 + 2 = with ≥ 0. Here > 0 is a parameter and the curve C is oriented positively if viewed from the positive ( > ) direction of the -axis. S First we note that the cylinder 2 ( /2) . Hence we can parameterize it by = = 2 2 + 2 sin 2 2 + = can be rewritten as ( − /2)2 + cos 0 ≤ < 2π 6 2 = We now substitute the above expressions for and to see that we have (as ≥ 0) √ 2√ = 1 − cos 2 in the equation 2 + 2 + 2 = 2 Thus, we have the following parametrization of Viviani’s curve: = + cos 2 = sin 2√ 2√ = 1 − cos 2 2 0 ≤ < 2π It’s also easy to see that the above parametrization gives the right orientation of the curve. The line integral now becomes 2 + 2 2 + C 2π √ 2√ 1 − cos 2 2 = + 0 2 + + cos 2 + cos + 2 2 √ 2√ 1 − cos 2 2 2 2 cos 2 2 sin Note that 2π 2 2 0 2 cos 2 √ 2√ 1 − cos 2 2π 0 2 cos 2 = 2 =− 0 + 2π 4 0 = 4 cos 3 2 3 2π =0 0 2π 3 sin 3 − cos 4 + 2 2π 3 = + 2 (1 − cos ) cos 0 1 + cos 2 2 π3 =− 4 Lastly, we have 2π 0 2 2 + 2 cos √ 2 2√ 1 − cos 7 √ 23 =− 16 2π...
View Full Document

## This document was uploaded on 02/10/2014.

Ask a homework question - tutors are online