∫∫∫Ω(x2+y2)dυ，其中Ω是由x2+y2=2z及z=2所围空间立体．
由于积分区域的柱面坐标表示为Ω={(θ,r，z)∣0≤θ≤2π，0≤r≤2，r2/2≤z≤2)} 所以∫∫∫Ω(x2+y2)dΩ=∫02πdθ∫02dr ∫(1/2)r22r2rdz =∫02πdθ∫02[r3•z∣(1/2)r22]dr =∫02πdθ∫02[2r3-(1/2)r5]dr =∫02π[(1/2)r4-(1/12)r6]∣02dθ =8/3∫02πdθ=(16/3)π
∫∫∫Ω(x2+y2)dυ，其中Ω是由x2+y2=2z及z=2所围空间立体．
由于积分区域的柱面坐标表示为Ω={(θ,r，z)∣0≤θ≤2π，0≤r≤2，r2/2≤z≤2)} 所以∫∫∫Ω(x2+y2)dΩ=∫02πdθ∫02dr ∫(1/2)r22r2rdz =∫02πdθ∫02[r3•z∣(1/2)r22]dr =∫02πdθ∫02[2r3-(1/2)r5]dr =∫02π[(1/2)r4-(1/12)r6]∣02dθ =8/3∫02πdθ=(16/3)π
