如何求旋转体的表面积与体积

Question

如题所述

Answer 1

先求面积：
S=∫[0--->1] (√x-x²) dx
=(2/3)x^(3/2)-(1/3)x³ |[0--->1]
=1/3
体积,显然绕x轴与绕y轴所得的旋转体体积是相同的,下面只求绕x轴的
V1=π∫[0--->1] (√x)² dx
=π∫[0--->1] x dx
=(π/2)x² |[0--->1]
=π/2
V2=π∫[0--->1] (x²)² dx
=π∫[0--->1] x⁴ dx
=(π/5)x⁵ |[0--->1]
=π/5
V=V1-V2=3π/10