Question

求摆线x=a(t-sin⁡t )， y=a(1- cos⁡t)， (0 ≤t≤2π) 绕x 轴和绕y 轴的旋转体体积

Answer 1

解：∵x=a(t-sint )， y=a(1- cost)， (0 ≤t≤2π)
∴dx=a(1-cost)dt
故 绕x轴的旋转体体积=∫<0,2πa>πy²dx
=π∫<0,2π>[a(1- cost)]²*a(1-cost)dt
=πa³∫<0,2π>[5/2-3cost+3cos(2t)/2-(1-sin²t)cost]dt
=πa³[5/2-3sint+3sin(2t)/4-(sint-sin³t/3)]│<0,2π>
=πa³[(5/2)(2π)
=5π²a³；
绕y轴的旋转体体积=∫<0,2πa>2πxydx
=2π∫<0,2π>a(t-sint )*a(1- cost)*a(1-cost)dt
=2πa³∫<0,2π>(t-sint )(1- cost)²dt
=2πa³[t(3t/2-2sint+sin(2t)/4)-(3t²/4+2cost-cos(2t)/4)-(1-cost)³/3]│<0,2π>
=2πa³(6π²-3π²)
=6π³a³。