求曲面z=1-4x²-y²和z=0所围立体的体积．

求曲面z=1-4x²-y²和z=0所围立体的体积．

V=4∫∫_D(1-4x²-y²)dxdy， 其中D={(x,y)∣0≤x≤1/2,0≤y≤√(1-4^x2)} 所以V=4∫₀^1/2dx∫₀^√1-4x2 (1-4x²-y²)dy =4∫₀^1/2[(1-4x²)y-(1/3)y³]∣ ₀^√1-4x2dx =8/3∫₀^1/2[√(1-4x²)³]dx =8/3∫₀^π/2cos³td[(1/2)sint]=4/3∫₀^π/2cos⁴tdt =1/3∫₀^π/2(1+cos2t)²dt=1/3∫₀^π/2(1+2cos2t+cos²2t)dt =π/6+1/3∫₀^π/2cos2td2t+1/6∫₀^π/2(1+cos4t)dt =π/6+(1/3)sin2t∣₀^π/2+π/12+1/24∫₀^π/2cos4td4t =π/4+(1/24)sin4t∣₀^π/2=π/4.