z=sqrt(x^2+y^2)
z=0
x2+y2=R^2
Полярные координаты:
x= ρ cos φ
y= ρ sin φ ⇒ x^2+y^2= ρ ^2 ⇒ [blue]sqrt(x^2+y^2)= ρ [/blue]
dxdy= [b]ρ d ρ d φ[/b]
0 ≤ ρ ≤ R
0 ≤ φ ≤ 2π
= ∫ ^(2π)_(0) ∫ ^(R)_(0) [blue]ρ [/blue]* [b]ρ d ρ d φ[/b]=∫ ^(2π)_(0) ([blue]∫ ^(R)_(0) ρ ^2d ρ[/blue] )d φ =∫ ^(2π)_(0) ( ρ ^3/3)|^(R)_(0)d φ =
∫ ^(2π)_(0)(( R ^3/3)-0)d φ =(R^3/3)∫ ^(2π)_(0)d φ =(R^3/3)*( φ )|^(2π)_(0)=(R^3/3)*(2π-0)=[red]2πR^3/3[/red]