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Unformatted text preview: 18.02 Exam 3
1 x2 0 Thursday, Nov 8, 2007 1:05  1:55 Problem 1. (15) Evaluate
0 xey dy dx by changing the order of integration. 1y Problem 2. (15) Find the average value x of the xcoordinate in the portion of the unit disk in the first quadrant enclosed by the yaxis and the line y = x. (Take the density = 1) Problem 3. (15) Using the change of variables u = xy, v = y 2  x2 , find the polar moment of inertia I0 of the region R enclosed by the yaxis, the line y = x, and the hyperbolas y 2  x2 = 1 and xy = 1 in the first quadrant (see picture); take the density = 1.
6 y 2 x 2 = 1 R y
xy = 1
 = x Problem 4. (20) a) (5) For which value of a is the vector field F = (3x2 2y sin x cos x)^+(a cos2 x+1)^ conservative? i b) (10) using the value of a you found in part (a), find a function f (x, y) such that F = f . (Use a systematic method. Show work.) c) (5) Let C be the right half of the unit circle, oriented counterclockwise, running from (0, 1) to (0, 1). Still using the value of a you found in part (a), calculate C F dr. Problem 5. (20) Let C1 be a line segment from (0, 0) to (1, 0), C2 an arc of the unit circle running from (1, 0) to (0, 1), and C3 a line segment from (0, 1) to (0, 0) (see figure). Let C be the simple closed curve formed by C1 , C2 , C3 , and let F = x3 ^ + x2 y ^. Calculate the line integral C F dr: i a) (10) directly;
C2 I C3 ? 0 C1 1 1 b) (10) using Green's theorem. Problem 6. (15) Consider the triangle with vertices P = (1, 0), Q = (1, 0) and R = (0, 1). For each of the two vector fields below, first determine without calculation the sign (positive, negative, zero) of the flux of F out of each side of the triangle, then calculate the total flux of F out of the triangle: a) (7) F = ^  ^; i b) (8) F = x^ + y^. i ...
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 Fall '08
 Auroux
 Vector Calculus, xey dy dx

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