计算上∫_Cyds，其中L为：x=a(t-sint)，y=a(1-cost)(0≤t≤2π)．

计算上∫_Cyds，其中L为：x=a(t-sint)，y=a(1-cost)(0≤t≤2π)．

因为dx/dt=a(1-cost)，dy/dt=asint ds=√[(dx/dt)²+(dy/dt)²ⁿ]dt=√[a²(1-cost)²+a²sin²]tdt =√2a√(1-cost)dt 所以∫_Cyds=∫₀^2πa(1-cost)•√2a√(1-cost)dt =√2a²∫₀^2π(1-cost)^3/2dt= √2a²∫₀^2π[2sin²(t/2)]^3/2)dt =4a²∫₀^2πsin²(t/2)dt=-8a²∫₀^2π [1-cos²(t/2)]dcos(t/2) =-8a²[cos(t/2)∣₀^2π-(1/3)cos³(t/2)∣₀^2π]=(32/3)a²