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

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MATH-4600 Homework VI Solutions to Graded Problems 1. 6.1 #1 f ( x, y ) = x + 2 y (a) r ( t ) = 2 - 3 t, 4 t - 1 , 0 t 2 C fds = b a f ( r ( t )) r ( t ) dt = 0 2 (2 - 3 t + 8 t - 2) - 3 , 4 dt = 0 2 (5 t )(5) dt = 25 2 t 2 2 0 = 50 (b) r ( t ) = cos ( t ) , sin ( t ) , 0 t π C fds = b a f ( r ( t )) r ( t ) dt = π 0 (cos( t ) + 2 sin( t )) - sin( t ) , cos( t ) dt = π 0 (cos( t ) + 2 sin( t )) dt = sin( t ) - 2 cos( t ) | π 0 = 4 2. 6.1 #2 f ( x, y, z ) = xyz, x ( t ) = t, 2 t, 3 t , 0 t 2 C fds = b a f ( r ( t )) r ( t ) dt = 2 0 6 t 3 1 , 2 , 3 dt = 2 0 6 t 3 14 dt = (6)(2 4 ) 14 4 = 24 14 1
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3. 6.1 #6 F ( x, y, z ) = x, y, z , x ( t ) = 2 t + 1 , t, 3 t - 1 , 0 t 1 C F · ds = b a f ( r ( t )) · r ( t ) dt = 1 0 2 t + 1 , t, 3 t - 1 · 2 , 1 , 3 dt = 1 0 (14 t - 1) dt = 7 t 2 - t 1 0 = 6 4. 6.1 #12 F ( x, y, z ) = 2 x, 1 , 0 , x ( t ) = t, 3 t 2 , 2 , 0 t 2 C F · ds = b a f ( r ( t )) · r ( t ) dt = 2 0 2 t, 1 , 0 · 1 , 6 t, 0 dt = 2 0 (2 t + 6 t ) dt = 4 t 2 2 0 = 16 5. 6.1 #13 r ( t ) = e 2 t cos(3 t ) , e 2 t sin(3 t ) , 0 t 2 π x xdx + ydy ( x 2 + y 2 )
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