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Unformatted text preview: Math 2374 Spring 2007 Midterm 2 Solutions  Page 1 of 4 March 28, 2007 1. (20 points) Evaluate 1 x x 2 (2 xy + x ) dy dx. Solution: 1 x x 2 (2 xy + x ) dy dx = 1 xy 2 + xy x x 2 dx = 1 ( x 3 + x 2 ) ( x 5 + x 3 ) dx = 1 x 2 x 5 dx = 1 3 x 3 1 6 x g 1 = 1 3 1 6 = 1 6 . There are two integrals that need to be evaluated in this problem, each one was worth 10 points. Correctly taking the antiderivative was 5 points, and correctly evaluating at the limits was 5 points. 2. (30 points) Consider the wire parametrized by r ( t ) = ( t cos t, t sin t, t ) for √ 2 ≤ t ≤ √ 7 . (i) (15 points) Set up, but do not evaluate, the integral for the length of the wire. r ( t ) = ( t sin t + cos t, t cos t + sin t, 1) 5 points l ( t ) = r ds = √ 7 √ 2  r ( t )  dt 5 points form; 1 point bounds l ( t ) = √ 7 √ 2 (cos t t sin t ) 2 + (sin t + t cos t ) 2 + 1 dt = √ 7 √ 2 √ 2 + t 2 dt 4 points (ii) (15 points) Suppose the density of the wire at the point ( x, y, z ) is δ ( x, y, z ) = z . Find the mass of the wire. (This should be an easy integral to calculate.) √ 7 √ 2 δ ( r ( t ))  r ( t )  dt 5 points √ 7 √ 2 t √ 2 + t 2 dt (2 points for t , 3 points for √ 2 + t 2 ) = 1 / 2 √ 7 √ 2 2 t √ 2 + t 2 dt = (1 / 2)(2 / 3)(2 + t 2 ) 3 / 2  √ 7 √ 2 = 1 / 3(9 3 / 2 4 3 / 2 ) = 19 / 3 5 points Notes: 2 for algebra mistakes 3. (20 points) Reverse the order of integration for this integral: 1 3 3 y (3 x y + 1) dx dy Solution:...
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 Spring '07
 Mosher
 Math, dxdy, ydx

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