计算三重积分,下标积分区域为Ω,求∫∫∫z^3dxdydz ,Ω为x^2+y^2+z^2≤1 ,z+1≥根号下x^2+y^2
问题描述：
计算三重积分,下标积分区域为Ω,求∫∫∫z^3dxdydz ,Ω为x^2+y^2+z^2≤1 ,z+1≥根号下x^2+y^2
答
原式=∫dθ∫rdr∫z³dz(作柱面坐标变换)
=(2π)(1/4)∫[(√(1-r²))^4-(r-1)^4]rdr
=(π/2)∫(4r^4-8r³+4r²)dr
=(π/2)[(4/5)r^5-2r^4+(4/3)r³]│
=(π/2)(4/5-2+4/3)
=(π/2)(2/15)
=π/15.