Question

∫(L)y^2ds，其中L为摆线的一拱x=a(t-sint)，y=(1-cost)，(0≦t≦2π，a＞0)

高等数学对弧长曲线积分

Answer 1

t：0→2π ds=√[(dx/dt)2+(dy/dt)2] dt=√[a2(1-cost)2+a2sin2t] dt=a√(2-2cost)dt=a√[4sin2(t/2)]dt=2asin(t/2)dt ∫ y ds =∫[0→2π] 2a2(1-cost)sin(t/2) dt =4a2∫[0→2π] sin3(t/2) dt =8a2∫[0→2π] sin3(t/2) d(t/2) =-8a2∫[0→2π] sin2(t/2) d[cos(t/2)] =-8a2∫[0→2π] [1-cos2(t/2)] d[cos(t/2)] =-8a2[cos(t/2) - (1/3)cos3(t/2)] |[0→2π] =-8a2(-1 - 1/3 - 1 - 1/3) =32a2/3