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prac3B

prac3B - f t b Evaluate the line integral using Green’s...

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18.02 Practice Exam 3 B Problem 1. a) Draw a picture of the region of integration of Z 1 0 Z 2 x x dydx. b) Exchange the order of integration to express the integral in part (a) in terms of integration in the order dxdy . Warning: your answer will have two pieces. Problem 2. a) Find the mass M of the upper half of the annulus 1 < x 2 + y 2 < 9 ( y 0) with density δ = y x 2 + y 2 . b) Express the x -coordinate of the center of mass, ¯ x , as an iterated integral. (Write explicitly the integrand and limits of integration.) Without evaluating the integral, explain why ¯ x = 0. Problem 3. a) Show that F = (3 x 2 - 6 y 2 ı + ( - 12 xy + 4 y is conservative. b) Find a potential function for F . c) Let C be the curve x = 1 + y 3 (1 - y ) 3 , 0 y 1. Calculate Z C F · d r . Problem 4. a) Express the work done by the force ±eld F = (5 x +3 y ı +(1+cos y on a particle moving counterclockwise once around the unit circle centered at the origin in the form Z b a f ( t ) dt . (Do not evaluate the integral; don’t even simplify
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Unformatted text preview: f ( t ).) b) Evaluate the line integral using Green’s theorem. Problem 5. Consider the rectangle R with vertices (0 , 0), (1 , 0), (1 , 4) and (0 , 4). The boundary of R is the curve C , consisting of C 1 , the segment from (0 , 0) to (1 , 0), C 2 , the segment from (1 , 0) to (1 , 4), C 3 the segment from (1 , 4) to (0 , 4) and C 4 the segment from (0 , 4) to (0 , 0). Consider the vector ±eld F = ( xy + sin x cos y )ˆ ı-(cos x sin y )ˆ a) Find the ²ux of F out of R through C . Show your reasoning. b) Is the total ²ux out of R through C 1 , C 2 and C 3 , more than, less than or equal to the ²ux out of R through C ? Show your reasoning. Problem 6. Find the volume of the region enclosed by the plane z = 4 and the surface z = (2 x-y ) 2 + ( x + y-1) 2 . (Suggestion: change of variables.)...
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