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Unformatted text preview: 18.02 Exam 3 Tuesday, Nov 4, 2008 1:05 - 1:55 Problem 1. (15) Evaluate integraldisplay 1 integraldisplay √ x- √ x 1 1 − y dy dx by changing the order of integration. Problem 2. (15) a) (5) Sketch the region R in the xy-plane corresponding to the iterated integral integraldisplay 1 integraldisplay √ 2- x 2 1 y x 2 + y 2 dy dx. b) (10) Set up the integral from part (a) in polar coordinates and evaluate it. Problem 3. (20) a) (5) For which value of a is the vector field vector F = (3 y 2 + axy +cos x )ˆ ı +3( x + y ) 2 ˆ conservative? b) (10) Using the value of a you found in part (a), find a function f ( x, y ) such that vector F = ∇ f . (Use a systematic method; show work) c) (5) Let C be the portion of the unit circle in the first quadrant, oriented counterclockwise, running from (1,0) to (0,1). Still using the value of a you found in part (a), calculate integraltext C vector F · dvector r . Problem 4. (20) Let C be the upper half of the unit circle, running counterclockwise from (1...
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This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
- Fall '08