求解：三重积分∫∫∫z^2dV,被积区域为x^2+y^2+z^2

Question

求解：三重积分∫∫∫z^2dV, 被积区域为x^2+y^2+z^2

Answer 1

用柱坐标做：x=rcosθ,y=rsinθ,z=z,dv=rdrdθdz,
∫∫∫z^2dV=
∫[-π/2,π/2]dθ∫[0,Rcosθ]rdr∫[-√(R^2-r^2),√(R^2-r^2)]z^2dz
=∫[-π/2,π/2]dθ∫[0,Rcosθ]rdr(2/3)*(R^2-r^2)^(3/2)
=∫[-π/2,π/2]dθ(2/15)*R^5*(1-(sinθ)^5)
=2πR^5/15.