Question

求摆线x=a(t-sint)y=a(1-cost)的一拱和x轴围成的图形绕x轴旋转所形成的旋转体的体积

Answer 1

解：所求体积=∫<0,2π>π[a(1-cosθ)]²*a(1-cosθ)dθ
=πa³∫<0,2π>(1-cosθ)³dθ
=πa³∫<0,2π>(1-3cosθ+3cos²θ-cos³θ)dθ
=πa³∫<0,2π>[5/2-3cosθ+(3/2)cos(2θ)-(1-sin²θ)cosθ]dθ
=πa³[5θ/2-3sinθ+(3/4)sin(2θ)-sinθ+sin³θ/3]│<0,2π>
=πa³[(5/2)(2π)]
=5π²a³