设D是Oxy平面上x=0，Y=0和a+y=1所围区域，求以D为底，曲面z=x²+y²为顶的曲顶柱体的体积．

设D是Oxy平面上x=0，Y=0和a+y=1所围区域，求以D为底，曲面z=x²+y²为顶的曲顶柱体的体积．

V=∫∫_D(x²+y²)dxdy =∫₀¹dx∫₀^1-x(x²+y²)dy= ∫₀¹[x²y+(1/3)y³)∣₀^1-xdx =∫₀¹x²(1-x)dx+(1/3)∫₀¹(1-x)³dx =[(1/3)x³∣₀¹]-[(1/4)x⁴∣₀¹]-1/12 (1-x)⁴∣₀¹=1/6