15_3_17 - t = 0 to t = T< π 2 is L = p b 2 c 2 Z T p 1-k...

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17. Helix: x = a cos t , y = b sin t , z = ct ( 0 < a < b ) . ds = p a 2 sin 2 t + b 2 cos 2 t + c 2 dt = p c 2 + b 2 - ( b 2 - a 2 ) sin 2 t dt = p b 2 + c 2 p 1 - k 2 sin 2 t dt ( k 2 = b 2 - a 2 b 2 + c 2 ). One complete revolution of the helix corresponds to 0 t 2 π , and has length L = p b 2 + c 2 Z 2 π 0 p 1 - k 2 sin 2 t dt = 4 p b 2 + c 2 Z π/ 2 0 p 1 - k 2 sin 2 t dt = 4 p b 2 + c 2 E ( k ) = 4 p b 2 + c 2 E s b 2 - a 2 b 2 + c 2 units . The length of the part of the helix from
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Unformatted text preview: t = 0 to t = T < π/ 2 is L = p b 2 + c 2 Z T p 1-k 2 sin 2 t dt = p b 2 + c 2 E ( k , T ) = p b 2 + c 2 E   s b 2-a 2 b 2 + c 2 , T   units....
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