exam3sol - 18.02 Exam 3 Tuesday Nov 4 2008 1:05 1:55...

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18.02 Exam 3 Tuesday, Nov 4, 2008 1:05 - 1:55 Problem 1. (15) Evaluate integraldisplay 1 0 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 0 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 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 .
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Problem 4. (20) Let C be the upper half of the unit circle, running counterclockwise from (1 , 0) to ( 1 , 0). Let vector F = (1 y 2 ı . Calculate the line integral integraltext C vector F · dvector r : a) (8) directly; b) (12) by applying Green’s theorem in a suitable manner.
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