Question

求圆x=a(cost+tsint),y=a(sint-tcost)的渐伸线的弧长

Answer 1

x'(t)=a(-sint+sint+tcost)=atcost,

y'(t)=a(cost-cost+tsint)=atsint,

x'(t)^2+y'(t)^2=a^2t^2,

s=∫ [0,2π] √ [x'(t)^2+y'(t)^2]dt

=∫ [0,2π] (at)dt

=a(t^2)/2[0,2π]

=2π^2a.

∴圆的渐开线一周弧长为2π^2a.