SpSolHw6 - MATH-4600 Homework VI Solutions to Graded...

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Unformatted text preview: MATH-4600 Homework VI Solutions to Graded Problems 1. 6.1 #1 f (x, y) = x + 2y (a) r(t) = 2 - 3t, 4t - 1 , 0 t 2 b f ds = C 0 a f (r(t)) r (t) dt = 2 (2 - 3t + 8t - 2) 0 -3, 4 2 dt = 2 (5t)(5)dt = 25 2 t 2 = 50 0 (b) r(t) = cos(t), sin(t) , 0 t b f ds = C a f (r(t)) r (t) dt = 0 (cos(t) + 2 sin(t)) - sin(t), cos(t) dt = 0 (cos(t) + 2 sin(t))dt = sin(t) - 2 cos(t)|0 = 4 2. 6.1 #2 f (x, y, z) = xyz, x(t) = t, 2t, 3t , 0 t 2 b f ds = C a 2 f (r(t)) 6t3 0 r (t) dt dt = 2 1, 2, 3 = 0 6t 3 (6)(24 ) 14 14dt = = 24 14 4 1 3. 6.1 #6 F (x, y, z) = x, y, z , x(t) = 2t + 1, t, 3t - 1 , 0 t 1 b F ds = C 1 a f (r(t)) r (t)dt = 0 2t + 1, t, 3t - 1 2, 1, 3 dt 1 = 0 (14t - 1)dt 1 0 = 7t2 - t 4. 6.1 #12 =6 F (x, y, z) = 2x, 1, 0 , x(t) = t, 3t2 , 2 , 0 t 2 b F ds = C 2 a f (r(t)) r (t)dt = 0 2t, 1, 0 1, 6t, 0 dt 2 = 0 (2t + 6t)dt 2 0 = 4t2 5. 6.1 #13 = 16 r(t) = e2t cos(3t), e2t sin(3t) , 0 t 2 xdx + ydy x b (x2 + y 2 ) 2 3 = a 2 x(t)x (t) + y(t)y (t) (x(t)2 + y(t)2 ) 2 3 dt dt = 0 e2t cos(3t)(2e2t cos(3t) - 3e2t sin 3t) + e2t sin(3t)(2e2t sin(3t) + 3e2t cos 3t) (e2t cos(3t))2 + (e2t sin(3t))2 ) 2 2 3 = 0 2e4t dt = e6t 2 0 2 e2t = 2 -e-2t 0 = 1 - e-4 2 ...
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SpSolHw6 - MATH-4600 Homework VI Solutions to Graded...

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