010.104-2007-2-fin - § ; ‡ „ ˚ ` ` ™ ¥ o...

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Unformatted text preview: § ; ‡ „ ˚ ` ` ™ ¥ o > ¢ „ 2 e K– § (2007 ‚ 12 Z 4 15 { 9 ‚˚ OE 1:00-3:00) < ˘ : s 2£ § : ¿ M % K V + I– Uc • ll ˙ a£ • ˆ Z k k ( ¥ @ \ 120 \ ). 1. (10 pts) Let c : [ a,b ] → R 2 be a unit speed parametrization of the path segment sitting between (0 , 0) and (1 , 1) of the parabola y = x 2 . Evaluate ∫ b a f ( c ( t )) dt , where f ( x,y ) = √ x 2 + 3 y + 1. 2. (10 pts) A parametrization for the hyperboloid of one sheet is given by x = (cosh u )(cos θ ) , y = (cosh u )(sin θ ) , z = sinh u, where 0 ≤ θ < 2 π and −∞ < u < ∞ . Find the tangent plane to this surface at the point ( x,y,z ) = (1 , 1 , 1). 3. (10 pts) For a vector field F ( x,y,z ) = y 3 i + (3 xy 2 − y 2 ) j and a simple path c ( t ) = (2 sin t, cos 3 t, 1 − cos 2 t ) (0 ≤ t ≤ π ), find the integral ∫ c F · d s . 4. (15 pts) Use Green’s Theorem to prove ∫ 2 π cos 2 n tdt = 2 n − 1 2 n ∫ 2 π cos 2 n − 2 tdt for any positive integer n ....
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This note was uploaded on 09/09/2009 for the course MATHMATICS 010-102 taught by Professor Cho during the Spring '09 term at Seoul National.

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