Задача 45349 ...

найти длину дуги

{ x = cos t + t sin t,
{ y = sin t – t cos t, oт t = 0 до t = π/4.

По формуле:

x`(t)=(cost+tsint)`=(cost)`+t`·sint+t·(sint)`=–sint+sint+t·cost=t·cost

y`(t)=(sint–tcost)`=(sint)`–t`·cost–t·(cost)`=cost–cost–t·(–sint)=t·sint

(x`(t))<sup>2</sup>+(y`(t))²=t²cos²t+t²sin²t=t²(sin²t+cos²t)=t²·1=t²

√(x`(t))<sup>2</sup>+(y`(t))²=t


L= ∫ ^π/4₀ tdt=(t²/2)|^π/4₀=π²/32