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Unformatted text preview: Math 241 Exam 4 Version 1 1. (a) (6pts) Evaluate Z r yz dx + xz dy + xy dz when r ( t ) = ( t,t 2 ,t 3 ) and t [0 , 3]. (b) (4pts) Give one of the physical interpretation of the line integral Z r F d s and use it to explain the effects of a reparametrization of the path r . 2. (a) (7pts) Evaluate ZZ z 3 dS where : [0 , 2 ] [0 , ] R 3 is parameterized cylinder of radius 2: ( s,t ) = (2cos s, 2sin s, 2 t ) . (b) (3pts) Let be a reparametrization of . Evaluate ZZ z 3 dS and briefly explain your answer. 3. Let ( s,t ) = (- s,s 2 + t,t 2 ). (a) (3pts) Find a normal for the surface described by . (b) (2pts) Are there any points in R 2 where this parametrization is not smooth? Justify your answer. (c) (5pts) Find an equation of the tangent plane to the surface at the point (0 , 1 , 1). 4. Let S be the surface defined by the portion of z = 4- x 2- y 2 above the xy-plane....
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