Задача 35802 ...
Условие
Решение
= ∫^(π/6)_(5π/6) 6sint*(6cost)`dt=
=-36 ∫^(π/6)_(5π/6) sin^2tdt=
=-36∫^(π/6)_(5π/6) (1-cos2t)dt/2=
=-18*(t-(1/2)sin2t)|^(π/6)_(5π/6)=
=18*(4π/6)+9*(sin(2π/6)-sin(10π/6))=
=12π+9*((sqrt(3)/2)-(-sqrt(3)/2))= [b]12π+9sqrt(3)[/b]
S=(1/2)∫^( β )_( α ) ρ^(2)( φ )d φ =
=6* ∫ ^(π/6)_(0)(3sin6 φ )^2d φ =
=54 ∫ ^(π/6)_(0)(sin^26 φ )d φ=
=54 ∫ ^(π/6)_(0)(1-cos12 φ )d φ/2=
=27*( φ - (1/12)sin(12 φ ))^(π/6)_(0)=
=27*(π/6)-(1/12)sin2π+(1/12)sin0=
[b]=9π/2[/b]
3.
x(t)=3*(2cost-cos2t)
y(t)=3*(2sint-sin2t)
L= ∫ ^(t_(2))_(t_(1))sqrt((x`(t))^2+(y`(t))^2)dt
x`(t)=3*(-2sint+2sin2t)
y`(t)=3*(2cost-2cos2t)
(x`(t))^2=9*(4sin^2t-8sint*sin2t+4sin^22t)
(y`(t))^2=9*(4cos^t-8costcos2t+4cos^22t)
(x`(t))^2+(y`(t))^2=9*(4*(sin^2t+cos^2t)-8sint*sin2t-8costcos2t+4*(sin^22t+cos^22t))=
=9*(8-8*(cos(2t-t))=9*8*(1-cost)=72*2sin^2(t/2)=144sin^2t/2
L= ∫ ^(2π)_(0)sqrt(144sin^2(t/2))dt=
=12 ∫ ^(2π)_(0)sin(t/2)dt=
=12(-2cos(t/2))^(2π)_(0)=
=-24*(cosπ-cos0)= [b]48[/b]
4.
L= ∫ ^( β )_( α )sqrt(ρ^2+(ρ`)^2)dφ
ρ=2e^(4φ /3)
ρ`=2e^(4 φ /3)*(4 φ /3)`=2*(4/3)*e^(4 φ /3)=(8/3)e^(4 φ /3)
ρ^2+(ρ`)^2=(2e^(4φ /3))^2+((8/3)e^(4φ /3))^2=
=(e^(4 φ /3))^2*(4+(64/9))= ((10/9)e^(4 φ /3))^2
L= ∫ ^( π/2 )_( - π/2)(10/9)*e^(4 φ /3)d φ =
=(3/4)*(10/9)e^(4 φ /3)|^(π/2)_(-π/2)=
= [b](5/6)*(e^(2π/3)-e^(-2π/3))[/b]