解：∵解方程组z=x²+2y²与z=6-2x²-y²，得x²+y²=2∴所求立体在xoy面上投影区域为D={(x，y)lx²+y²≤2}故 所求立体体积=∫∫[(6-2x²-y²)-(x²+2y²)]dxdy=∫∫[6-3(x²+y²)]dxdy=∫<0,2π>dθ∫<0,√2>(6-3r²)rdr (应用极坐标变换)=2π∫<0,√2>(6r²-3r³)dr=2π(2r³-3r^4/4)│<0,√2>=2π(4√2-3)
