6 1 1 work done c f dr 6 0 5 7 t t t 2 3

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Unformatted text preview: mass of the wire is 2r2 and the center of mass is 1 Note: sin t cos t = 2 sin 2t, sin2 t = substitution, let u = nt. 1−cos 2t 2 and cos2 t = 5. r (t) = [t, t2 ], −1 ≤ t ≤ 2 F • dr = C 2 For sin nt and cos nt, use −1 t sin t2 , t2 • [1, 2t] dt t sin t2 + 2t3 dt = = −1 Note: for nt 1+cos 2t . 2 . 2 Work done = n−1 . 0 C My = 4 ,0 π π 2 4. C = x (t) = r cos t, y (t) = r sin t, 0 ≤ t ≤ π 2 = 15 + cos 1 − cos 4 2 n sin t , use substitution, let u = tn . 6. 1 1 Work done = C F • dr = 6 0 5 7 t , −t , −t 2 3 • 2t, −3t , 4t dt = 0 The Fundamental Theorem for Line Integrals 1. (a) x2 + g (y ) 2 y2 =⇒fy (x, y ) = g ′ (y ) = y =⇒ g (y ) = +C 2 x2 + y 2 =⇒f (x, y ) = +C 2 f (x, y ) = fx dx = 5 x dx = 5t7 − 4t10 dt = 23 88 C F • dr = f (x1 , y1 ) − f (x0 , y0 ) = f (3, 9) − f (−1, 1) = 45 − 1 = 44 Note: if we evaluate the line integral directly, then C = {x (t) = t, y (t) = t2 , − 1 ≤ t ≤ 3} and 3 3 C F • dr = 2 t, t −1 • [1, 2t] dt = t + 2t3 dt = 44. −1 (b) f (x, y ) = x2 y 3 + C π F • dr = f r 2 C π2 + 4 1, 4 − f (r (0)) = f 3 (π 2 + 4) − f (0, 1) = 64 (c) f (x, y, z ) = x2 y 3 z 4 + C C F • dr = f (r (2)) − f (r (0)) = f (2, 4, 8) − f (0, 0, 0) = 220 (d) f (x, y, z ) = x2 z + x sin y + C C F • dr = f (r (2π )) − f (r (0)) = f (1, 0, 2π ) − f (1, 0, 0) = 2π 2. (a) C 2x sin y dx + x2 cos y − 3y 2 dy = C 2x sin y, x2 cos y − 3y 2 • [dx, dy ] So, F (x, y ) = 2x sin y, x2 cos y − 3y 2 . curl F = i j k ∂ ∂ ∂ ∂x ∂y ∂z 2x sin y x2 cos y − 3y 2 0 = [0 − 0, 0 − 0, 2x cos y − 2x cos y ] = 0 Hence, F is conservative and the line integral is independent of path. The potential function f (x, y ) exists and f (x, y ) = x2 sin y − y 3 + C . C 2x sin y dx + x2 cos y − 3y 2 dy = f (5, 1) − f (−1, 0) = 25 sin 1 − 1 (b) F = 2y 2 − 12x3 y 3 , 4xy − 9x4 y 2 curl F = 0 − 0, 0 − 0, 4y − 36x3 y 2 − 4y + 36x3 y 2 = 0 Hence, F is conservative and the line integral is independent of path. The potential function f (x, y ) exists and f (x, y ) = 2xy 2 − 3x4 y 3 + C . C 2y 2 − 12x3 y 3 dx + 4xy − 9x4 y 2 dy = f (3, 2) − f (1, 1) = −1919 6...
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This document was uploaded on 01/23/2014.

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