Question

Вычислите дуги кривой. вышмат. С решением

Answer 1

$x'_t=3cos^2tcdot(-sint)\ y'_t=3sin^2tcdot cost\ L=intlimits_0^pi sqrt{(3cos^2tcdot(-sint))^2+(3sin^2tcdot cost)^2}dt=3intlimits_0^pi sqrt{cos^4tcdot sin^2t+sin^4tcdot cos^2t}dt=3intlimits_0^pi sqrt{cos^2tcdot sin^2t(cos^2t+sin^2t)}dt=3intlimits_0^pi sqrt{cos^2tcdot sin^2t}dt=3intlimits_0^{frac{pi}{2}} sqrt{cos^2tcdot sin^2t}dt+3intlimits_{frac{pi}{2}}^pi sqrt{cos^2tcdot sin^2t}dt=$$=3intlimits_0^{frac{pi}{2}} costcdot sintdt-3intlimits_{frac{pi}{2}}^pi costcdot sint dt=dfrac{3}{4}intlimits_0^{frac{pi}{2}} sin2td(2t)-dfrac{3}{4}intlimits_{frac{pi}{2}}^pi sin2t d(2t)=dfrac{3}{4}((-cos2t)|_0^{frac{pi}{2}}-(-cos2t)|_{frac{pi}{2}}^pi)=dfrac{3}{4}((1+1)-(-1-1))=3$