求由曲面z=x^2+y^2,z=4-y^2所围立体的体积,用三重积分

Question

求由曲面z=x^2+y^2,z=4-y^2所围立体的体积,用三重积分

Answer 1

∵所求体积在xy平面的投影是S：x²/4+y²/2=1
∴所求体积=∫∫[(4-y²)-(x²+y²)]dxdy
=∫∫(4-x²-2y²)dxdy
=∫dθ∫(4-4r²)*2√2rdr (做坐标变换x=2rcosθ,y=√2rsinθ)
=16√2π∫(r-r³)dr
=16√2π(r²/2-r^4/4)│
=16√2π(1/2-1/4)
=4√2π.