∫Ly²ds,其中L是摆线的一拱x＝a(t－sint),y＝a(1－cost),(0≤t≤2л)

Question

∫Ly²ds,其中L是摆线的一拱x＝a(t－sint),y＝a(1－cost),(0≤t≤2л)

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