x= ρ cos θ
y= ρ sin θ
x^2+y^2= ρ^2
Якобиан |J|= ρ
x^2+y^2=π^2 ⇒ ρ ^2=π^2 ⇒ ρ =π
x^2+y^2=4π^2⇒ ρ ^2=4π^2 ⇒ ρ =2π
D:
π ≤ ρ ≤ 2π
0 ≤ θ ≤ 2π
∫ ∫ _(D)(sin ρ )* ρ d ρ d θ = ∫ ^(2π)_(0)[b]([/b] ∫ ^(2π)_(π) ρ *sin ρ d ρ[b])[/b] d θ=
считаем внутренний интеграл по частям:
u= ρ dv=sin ρ d ρ
du=d ρ ; v=-cos ρ
= ∫ ^(2π)_(0)[b]([/b](- ρ cos ρ )|^(2π)_(π)+ ∫^(2π)_(π) cos ρ d ρ [b])[/b]d θ =
= ∫ ^(2π)_(0)[b]([/b] -(2π)*cos2π+π*cosπ+sin ρ |^(2π)_(π)[b])[/b]d θ =
=∫ ^(2π)_(0)[b]([/b](-3π+0)d θ =
=-3π θ |^(2π)_(0)=-[b]6π[/b]