010.104-2008-2-final - Honor Calculus II Final Exam...

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Unformatted text preview: Honor Calculus II Final Exam (December 6, 2008. 13:00-15:00) < : s 2 : M % K V + I U c l a Z k k . ( @ 200 ) 1. (20 pts) Calculate the following integral Z 3 Z 9 x 2 x 3 e y 3 dydx. 2. (20 pts) Let S be the surface defined by ρ = 1- cos ϕ in the spherical coordinates. Find the volume of the region enclosed by the surface S . Find the center ( x, y, z ) of the region. 3. (20 pts) Using Green’s theorem, evaluate the following integral Z C ( x 3 + y cos xy ) dx + ( y 3 + x cos xy ) dy along the curve C : x 2 + 4 y 2 = 4 (0 ≤ x ≤ 2 , y ≥ 0) from (2 , 0) to (0 , 1). 4. (20 pts) A plane at a distance of 2 from the origin (0 , , 0) divides the ball of radius 3 centered at the origin into two parts. Find the solid angle of the smaller piece at the origin. (The solid angle subtended by some region at a point P is defined as the area of the projection of that region onto a unit sphere centered at P .) 5. (30 pts) For a C 2 vector field F ( x, y ) = P ( x, y ) i + Q ( x, y ) j defined on an open set...
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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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