求由平面x=0，y=0，x+y=1所围成的柱体被平面z=0及抛物面x2+y2=6-z截得的立体的体积．
由题意知：V=∫∫D(6-x2-y2)dxdy =∫01dx∫01-x(6-x2-y2)dy =∫01[6y-x2y-y3/3∣01-x]dx =∫01[6(1-x)-x2(1-x)-(1-x)3/3]dx =∫01(1-x)[-(4/3)x2+(2/3)x+17/3]dx=17/6．
求由平面x=0，y=0，x+y=1所围成的柱体被平面z=0及抛物面x2+y2=6-z截得的立体的体积．
由题意知：V=∫∫D(6-x2-y2)dxdy =∫01dx∫01-x(6-x2-y2)dy =∫01[6y-x2y-y3/3∣01-x]dx =∫01[6(1-x)-x2(1-x)-(1-x)3/3]dx =∫01(1-x)[-(4/3)x2+(2/3)x+17/3]dx=17/6．
