两道定积分求旋转体体积的问题，希望的学哥学姐老师们帮帮忙，给出一个详细的解答，谢谢！

Question

如题所述

Answer 1

y = sinx，0 ≤ x ≤ π
绕x轴：
V = πy²
= π∫[0→π] sin²x dx
= (π/2)∫[0→π] (1 - cos2x) dx
= (π/2)[x - (1/2)sin2x] |[0→π]
= (π/2)(π)
= π²/2
绕y轴：
V = 2πxy
= 2π∫[0→π] xsinx dx
= - 2π∫[0→π] x d(cosx)
= - 2π[xcosx] |[0→π] + 2π∫[0→π] cosx dx
= - 2π[- π] + [sinx] |[0→π]
= 2π²

y = x(x - 1)，y = 0，x = 2，1 ≤ x ≤ 2
A = ∫[1→2] x(x - 1) dx
= ∫[1→2] (x² - x) dx
= [x³/3 - x²/2] |[1→2]
= [8/3 - 4/2] - [1/3 - 1/2]
= 5/6
V = 2πxy
= 2π∫[1→2] x[x(x - 1)] dx
= 2π∫[1→2] (x³ - x²) dx
= 2π[x⁴/4 - x³/3] |[1→2]
= 2π{[16/4 - 8/3] - [1/4 - 1/3]}
= 17π/6

Answer 2

定居分