曲線y=根號x-1,y=x/2,與x軸圍成的平面圖形繞x軸y軸旋轉一周所得的體積是多少?（用定積分來求）,

數(shù)學人氣：121 ℃時間：2019-08-19 01:57:05

優(yōu)質解答

繞x軸旋轉一周所得的體積=∫π(x²/4)dx-∫π(x-1)dx
=[(π/12)x³]│-[π(x²/2-x)]│
=(π/12)(2³-0³)-π(2²/2-2-1²/2+1)
=2π/3-π/2
=π/6；
繞y軸旋轉一周所得的體積=∫2πx(x/2)dx-∫2πx√(x-1)dx
=π∫x²dx-2π∫[(x-1)^(3/2)+(x-1)^(1/2)]dx
=[π(x³/3)]│-2π[(2/5)(x-1)^(5/2)+(2/3)(x-1)^(3/2)]│
=(π/3)(2³-0³)-2π[(2/5)(2-1)^(5/2)+(2/3)(2-1)^(3/2)]
=8π/3-32π/15
=8π/15.