15_5_20 - 20. x = au cos v, y = au sin v, z = bv, (0 u 1, 0...

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20. x = au cos v, y = au sin v, z = b v , ( 0 u 1 , 0 v 2 π) . This surface is a spiral (helical) ramp of radius a and height 2 π b , wound around the z -axis. (It’s like a circular staircase with a ramp instead of stairs.) We have ∂( x , y ) ∂( u , v) = ± ± ± ± a cos v - au sin v a sin v au cos v ± ± ± ± = a 2 u ∂( y , z ) ∂( u , v) = ± ± ± ± a sin v au cos v 0 b ± ± ± ± = ab sin v ∂( z , x ) ∂( u , v) = ± ± ± ± 0 b a cos v - au sin v ± ± ± ± = - ab cos v dS = p a 4 u 2 + a 2 b 2 sin 2 v + a 2 b 2 cos 2 v du d v = a p a 2 u 2 + b 2 du d v. The area of the ramp is A = a Z 1 0 p a 2 u 2 + b 2 du Z 2 π 0 d v = 2 π a Z 1 0 p a 2 u 2 + b 2 du Let au = b tan θ a du = b sec 2 θ d θ = 2 π b 2 Z u = 1
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