Question

用球面积分法解这道三重积分

高等数学

用球面积分法解这道三重积分

Answer 1

Answer 2

解：原式=∫<0,2π>dθ∫<0,√3R/2>rdr∫<R-√(R²-r²),√(R²-r²)>z²dz (作柱面坐标变换)
=(2π/3)∫<0,√3R/2>[(√(R²-r²))³-(R-√(R²-r²))³]rdr
=(2π/3)*(-1/2)∫<0,√3R/2>[(2√(R²-r²)³-3R(R²-r²)+3R²√(R²-r²)-R³]d(R²-r²)
=(-π/3)[(4/5)(R²-r²)^(5/2)-(3R/2)(R²-r²)²+2R²(R²-r²)^(3/2)-R³(R²-r²)]│<0,√3R/2>
=(-π/3)[(4/5)(R/2)^5-(3R/2)(R/2)^4+2R²(R/2)³-R³(R/2)²-(4/5)R^5+(3R/2)R^4-2R²*R³+R³*R²]
=(-π/3)(-59/160)R^5
=(59π/480)R^5。