極坐標(biāo)標(biāo)
∫∫ √(R²-x²-y²) dxdy
=∫∫ r√(R²-r²) drdθ
=∫[-π/2→π/2] dθ∫[0→Rcosθ] r√(R²-r²) dr
=(1/2)∫[-π/2→π/2] dθ∫[0→Rcosθ] √(R²-r²) d(r²)
=-(1/2)(2/3)∫[-π/2→π/2] (R²-r²)^(3/2) |[0→Rcosθ] dθ
=(1/3)∫[-π/2→π/2] (R³-R³|sinθ|³) dθ
=(2R³/3)∫[0→π/2] (1-sin³θ) dθ
=(2R³/3)[∫[0→π/2] 1 dθ - ∫[0→π/2] sin³θ dθ]
=(2R³/3)[π/2 + ∫[0→π/2] sin²θ d(cosθ)]
=(2R³/3)[π/2 + ∫[0→π/2] (1-cos²θ) d(cosθ)]
=(2R³/3)[π/2 + θ - (1/3)cos³θ] |[0→π/2]
=(2R³/3)[π/2 + π/2 - 0]
=2πR³/3
若有不懂請(qǐng)追問(wèn),如果解決問(wèn)題請(qǐng)點(diǎn)下面的“選為滿(mǎn)意答案”.你錯(cuò)了，答案是1/3R^3(π-4/3)倒數(shù)第三步時(shí)出錯(cuò)了，從倒數(shù)第四步開(kāi)始=(2R³/3)[π/2 + ∫[0→π/2] (1-cos²θ) d(cosθ)]=(2R³/3)[π/2 + cosθ - (1/3)cos³θ]|[0→π/2]=(2R³/3)[π/2 -1 + 1/3]=(2R³/3)[π/2 - 2/3]=(R³/3)[π - 4/3]