Find the length of the cycloid
x(t)=4(t−sin⁡(t))\displaystyle {x}{\left({t}\right)}={4}{\left({t}-{\sin{{\left({t}\right)}}}\right)}
y(t)=4(1−cos⁡(t))\displaystyle {y}{\left({t}\right)}={4}{\left({1}-{\cos{{\left({t}\right)}}}\right)}
for 0≤t≤2π\displaystyle {0}\le{t}\le{2}\pi

cycloid from Hoffman's Contemporary Calculus

The length is exactly  

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