Задача 35883 ...
Условие
Решение
y=x^2/5 - парабола красным цветом
y=sqrt(5x) - степенная функция, синим
S= ∫ ^(5)_(0)(sqrt(5x)-(x^2/5))dx=
= [b]([/b]sqrt(5)*x^(3/2)/(3/2)-(1/5)*(x^3/3) [b])[/b]|^(5)_(0)=
=(2sqrt(5)/3)*sqrt(5^3)-(1/15)*5^(3)=
=(50/3)-(25/3)= [b]25/3[/b]
б)
S= ∫ ^(t_(2))_(t_(1))y(t)*x`(t)dt
y(t)=8cos^3t
x(t)=2sin^3t
x`(t)=2*3*sin^2t*(sint)`=6sin^2tcost
S= ∫ ^(π/4)_(0)8*cos^3t*6sin^2tcostdt=
=48 ∫ ^(π/4)_(0)cos^4t*sin^2tdt= понижаем степени
=48 ∫ ^(π/4)_(0)((1+cos2t)/2)^2*(1-cos2t)dt/2=
=6 ∫ ^(π/4)_(0)((1-cos^22t)*(1+cos2t)dt=
=6 ∫ ^(π/4)_(0)((1-cos^22t+cos2t-cos^32t)dt=
=6 ∫ ^(π/4)_(0)((1-(1+cos4t)/2+cos2t-(1-sin^22t)*cos2t)dt=
=6* [b]([/b](t/2)-(1/8)sin4t +(1/2)(sin^32t)/3 [b])[/b]|^(π/4)_(0)=
=6π/8= [b]3π/4[/b]
в)
S=(1/2) ∫ ^( β )_( α )ρ^(2)( φ )d φ
cos4φ ≥0
-π/2 ≤4φ≤π/2
-π/8 ≤φ≤π/8
4 лепестка ( см. рис.2),
считаем площадь половины лепестка и умножаем на 8
S=(1/2) *8*∫ ^(π/8)_(0 )(2cos4φ)^2dφ=
=16*∫^(π/8)_(0 )(cos^24φ)dφ= понижаем степень
=16*∫ ^(π/8)_(0) ((1+сos8φ)/2)dφ=
=8*( φ +(1/8)sin8 φ )|^(π/8)_(0) =
=8*(π/8)= [b]π[/b]
