设积分区域Ω由上半球面z=√1-x2-y2及平面z=0所围成，求三重积分∫∫∫Ωzdxdydz．
∫∫∫Ωzdxdydz=∫∫x2+y2≤1(∫0√1-x2-y2zdz)dxdy =∫∫x2+y2≤11/2(1-x2-y2)dxdy =∫02πdθ∫011/2(1-r2)rdr =π[(1/2)r2-(1/4)r4]∣01=π/4.
设积分区域Ω由上半球面z=√1-x2-y2及平面z=0所围成，求三重积分∫∫∫Ωzdxdydz．
∫∫∫Ωzdxdydz=∫∫x2+y2≤1(∫0√1-x2-y2zdz)dxdy =∫∫x2+y2≤11/2(1-x2-y2)dxdy =∫02πdθ∫011/2(1-r2)rdr =π[(1/2)r2-(1/4)r4]∣01=π/4.
