Here 0 is a parameter and the curve c is oriented

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

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