Question

设曲线a=acost,y=bsint,a>b,0<=t<=2π在M(x,y)处的线密度等于ρ=|y|,求其质量！答案如下，求大神详解！！

答案为2b(b+a*arcsinε/ε），ε为椭圆的离心率

Answer 1

dx/dt=-a*sint,dy/dt=b*cost,
ds=√((dx/dt)^2+(dy/dt)^2)dt=√(a^2*sint^2+b^2*cost^2)dt
其质量=∫ρds=4∫(0,π/2)b*sint*√(a^2*sint^2+b^2*cost^2)dt
=4∫(0,π/2)b*sint*√(a^2-（a^2-b^2）*cost^2)dt
=4∫(0,π/2)b*sint*√(a^2-c^2*cost^2)dt
=4∫(0,π/2)a*b*sint√(1-e^2*cost^2)dt
=4 a*b*∫(0,π/2) sint√(1-e^2*cost^2)dt
=4 a*b*[1/2*((1-e^2)^(1/2)*e+arcsin(e))/e]
=2 a*b*(b/a+1/e*arcsin(e))
=2 b*(b+a/e*arcsin(e))