Question

计算I=三重积分z^2dxdydz,其中积分区域为x^2+y^2+z^2

Answer 1

简单计算一下即可，答案如图所示

Answer 2

原式=∫dθ∫rdr∫z²dz (做柱面坐标变换)
=(2π/3)∫[(a²-r²)^(3/2)-(a-√(a²-r²))³]rdr
=(π/3)∫[a³-3a²(a²-r²)^(1/2)+3a(a²-r²)-2(a²-r²)^(3/2)]d(a²-r²)
=(π/3)[a³(a²-r²)-2a²(a²-r²)^(3/2)+(3/2)a(a²-r²)²-(4/5)(a²-r²)^(5/2)]│
=(π/3)(a^5/4-a^5/4+3a^5/32-a^5/40-a^5+2a^5-3a^5/2+4a^5/5)
=59π/480。