∵所围成图形是关于xz平面和yz平面对称的
∴所求体积=4×第一卦限体积
∵由x²+y²+z²=R²==>z=√(R²-x²-y²)
由x²+y²+z²=2Rz==>z=R-√(R²-x²-y²)
∴第一卦限体积是由曲面z=√(R²-x²-y²)与z=R-√(R²-x²-y²),以及xz平面和yz平面(x,y>0)所围成
∵由x²+y²+z²=R²与x²+y²+z²=2Rz解方程,得x²+y²=(√3R/2)²
∴所求体积在xy平面的投影是圆x²+y²=(√3R/2)²
故所求体积=4×第一卦限体积
=4∫∫{√(R²-x²-y²)-[R-√(R²-x²-y²)]}dxdy
=4∫∫[2√(R²-x²-y²)-R]dxdy
=4∫dθ∫[2√(R²-ρ²)-R]ρdρ (极坐标变换)
=π∫[2√(R²-ρ²)-R]d(ρ²)
=π[(-4/3)(R²-ρ²)^(3/2)-Rρ²]│
=π[(-4/3)(R²-3R²/4)^(3/2)-3R²/4+(4/3)(R²-0)^(3/2)+R*0]
=π(-R³/6-3R³/4+4R³/3)
=5πR³/12.