Задача 27839 x(t)=2*(cost-t*sint), ...

u7711124432

22.05.2018

x(t)=2*(cost-t*sint), y(t)=2*(sint-t*cost) 0 < t < pi/2 Найти длину дуги

SOVA

27.05.2018

x`(t)=2*(cost[b]+[/b]tsint)`=2*(-sint+sint+tcost)=2tcost
y`(t)=2(sint-tcost)`=2*(cost-cost+tsint)=2tsint

(x`(t))^2=4t^2cos^2t)
(y`(t))^2=4t^2sin^2t

(x`(t))^2+(y`(t))^2=4t^2(cos^2t+sin^2t)=4t^2

L= ∫ ^(Pi/2)_(0)sqrt(4t^2)dt=(2t^2/2)|^(Pi/2)_(0)=Pi^2/4
О т в е т. Pi^2/4