极坐标求∫∫根号下(x^2+y^2)dxdy,其中d是由x^2+Y^2>=ay,X^2+Y^2

Question

极坐标求∫∫根号下(x^2+y^2)dxdy,其中d是由x^2+Y^2>=ay,X^2+Y^2

Answer 1

∫∫ √(x² + y²) dxdy
= ∫∫D₁ - ∫∫D₂
= ∫(0→2π) dθ ∫(0→a) r² dr - ∫(0→π) dθ ∫(0→asinθ) r² dr
= 2π[ r³/3 ]：(0→a) - ∫(0→π) [ r³/3 ]：(0→asinθ) dθ
= 2π * a³/3 - (a³/3)(2)∫(0→π/2) sin³θ dθ
= (2/3)πa³ - (2/3)a³ * 2/(3 * 1)
= (2/3)πa³ - (4/9)a³
= (2π/3 - 4/9)a³