dfgdggdg - where F x y z = xz i yz j z 2 k and Σ is the...

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MATH 241H - Exam 4 - May 7, 2007 Show your work. No calculators. [15] 1. Evaluate R C y sin z dx + (2 y + x sin z ) dy + xy cos z dz where C is the curve param- eterized by r ( t ) = cos( πt ) i + e t j + πt 3 / 2 k , 0 t 1. [15] 2. Evaluate R C ( xy + z ) ds where C is the curve parameterized by r ( t ) = t i + 3 t j + (2 t + 3) k , 0 t 1 . [15] 3. Evaluate R C ( x 2 + 2 y ) dx + sin( y 3 ) dy where C is the curve shown below consisting of the part of the x -axis between ( - 1 , 0) and (1 , 0) and the part of the parabola y = 1 - x 2 above the x axis. [20] 4. Let D be the solid region bounded below by the cone z = p x 2 + y 2 and bounded above by the plane z = 3. Evaluate R R Σ F · n dS
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Unformatted text preview: where F ( x, y, z ) = xz i + yz j + z 2 k and Σ is the entire surface of D with inward pointing normal vectors. 5. For both parts (a) and (b), let Σ be the part of the plane z = 5-x-3 y for which ≤ x ≤ 1 , ≤ y ≤ 2. [15] (a) Evaluate R R Σ z dS . [20] (b) Evaluate R C F · d r where F ( x, y, z ) = ( x 2 + yz ) i + (sin y + x ) j + ( e z + y ) k and C is the entire boundary of Σ, oriented counterclockwise viewed from above....
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