曲线x=at,y=1/2at^2,z=1/3at^3(a>0,0<=t<=1),线密度u=(2y/a)^0.5
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解决时间 2021-04-07 08:42
- 提问者网友：难遇难求
- 2021-04-07 03:02
曲线x=at,y=1/2at^2,z=1/3at^3(a>0,0<=t<=1),线密度u=(2y/a)^0.5
最佳答案
- 五星知识达人网友：迷人又混蛋
- 2021-04-07 03:57
曲线的线密度 u=t
质量
m =∫{0,1}tds = ∫{0,1}t(a^2+(at)^2+(at^2)^2)^(1/2)dt
=（a/2）∫{0,1}(1+u+u^2)^(1/2)du{这里u=t^2}
=(a/4)*[(u+1/2)(u^2+u+1)^(1/2)+(3/4)*ln(u+1/2+(u^2+u+1)^1/2)]_{0,1}
=(a/4)*[(3/2)*√3 +(3/4)*ln((3/2)+√3) -(1/2)-(3/4)*ln(3/2) ]
质量
m =∫{0,1}tds = ∫{0,1}t(a^2+(at)^2+(at^2)^2)^(1/2)dt
=（a/2）∫{0,1}(1+u+u^2)^(1/2)du{这里u=t^2}
=(a/4)*[(u+1/2)(u^2+u+1)^(1/2)+(3/4)*ln(u+1/2+(u^2+u+1)^1/2)]_{0,1}
=(a/4)*[(3/2)*√3 +(3/4)*ln((3/2)+√3) -(1/2)-(3/4)*ln(3/2) ]
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