求摆线x=a(t-sint ), y=a(1- cost), (0 ≤t≤2π) 绕x 轴...
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发布时间:2024-10-07 08:20
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时间:2024-10-08 08:36
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。