The cycloid curve - Section 9.2

ParametricPlot[{t-Sin[t],1-Cos[t]},{t,-2Pi,6Pi},ImageSize->700]

ParametricPlot[{t-Sin[t],1-Cos[t]},{t,0,2Pi},ImageSize->700]

Integrate[Sqrt[D[t-Sin[t],t]^2+D[1-Cos[t],t]^2],{t,0,2Pi}]