m251ex3key

# m251ex3key - A~ MATH 251 I<~ Section 504 Test III April 7...

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A~ I<~ MATH 251 Section 504 Test III April 7, 2004 Name There are a total of 5 problems. 1. Given the circular helix x(t) = acos2t } y(t) = asin2t z(t) = bt 0 .s t .s 311"; a, b > 0, (i) Let s(t) be the arclength variable of the circular helix: 1. 2. 3. 4. 5.- Total s = s(t) = It ds. Determine s(t) as a functionof t. (7.5%) ...... (ii) Determine the unit tangent vector T(t) on the curve at time t. (7.5%) (Hi) Evaluate the line integral J y sin z ds, where b = 2 while a > 0 is arbitrary, by C using the s variable. (iv) Repeat part (iii) above, but by using the t variable. ( i) (+ .4 Stt)= Jt> h'(~lt:{z ~, ( Y (i):: -Zo.. ~7.,t) 2..a.~1..t ) h) l -a. I ( 1. -a.- a. J Yl... ?(-c) = (-LA ~2...t) +-C2..0.. ~:z..t) +- b ::: (iftt2.(~l,-t+-~t,t) +-b3-J y~ (4tt2-+b1.J Yz. ~tt:);:. ); (447..+ bt.y\:(z;;~ ~o.."1.+hi. + 1 "-~i I 7t~<.I :::- (7.5%) (7.5%) -

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