∫∫ρ^4(cosθ)^2(sinθ)^2√(a^2-ρ^2)*ρdρdθ,0<=ρ<=a,0<=θ<=2π

∫∫ρ^4(cosθ)^2(sinθ)^2√(a^2-ρ^2)*ρdρdθ,0<=ρ<=a,0<=θ<=2π

∫∫ ρ⁴cos²θsin²θ√(a² - ρ²) · ρdρdθ，0 ≤ ρ ≤ a，0 ≤ θ ≤ 2π
= ∫(0→2π) cos²θsin²θ dθ ∫(0→a) ρ⁴√(a² - ρ²) · ρdρ
= ∫(0→2π) cos²θsin²θ dθ ∫(0→a) (- ρ²)²√(a² - ρ²) d(ρ²/2)
= ∫(0→2π) cos²θsin²θ dθ ∫(0→a) (- 1/2)[(a² - ρ²) - a²]²√(a² - ρ²) d(a² - ρ²)
= (- 1/2)∫(0→2π) cos²θsin²θ dθ ∫(0→a) [(a² - ρ²)^(5/2) - 2a²(a² - ρ²)^(3/2) + a⁴√(a² - ρ²)] d(a² - ρ²)
= (- 1/2)[(2/7)(a² - ρ²)^(7/2) - (4/5)a²(a² - ρ²)^(5/2) + (2a⁴/3)(a² - ρ²)^(3/2)]∫(0→2π) cos²θsin²θ dθ
= (- 1/105)(8a⁴ + 12a²ρ² + 15ρ⁴)(a² - ρ²)^(3/2) · ∫(0→2π) (1/2 · sin(2θ))² dθ
= 8a⁷/105 · (1/4)∫(0→2π) sin²(2θ) dθ
= 8a⁷/105 · (1/4)(1/2)∫(0→2π) (1 - cos(4θ)) dθ
= 8a⁷/105 · (1/8)(θ - (1/4)sin4θ)：[0→2π]
= 8a⁷/105 · (1/8)(2π)
= 2πa⁷/105

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