What is the length of the curve with parametric equations x = t - sin(t), y = 1 - cos(t) from t = 0 to t = 2π?

Question

What is the length of the curve with parametric equations x = t - sin(t), y = 1 - cos(t) from t = 0 to

t = 2π?

Answer 1

S(t) = ∫√((dx/dt)² + (dy/dt)²)dt from a to b

dx/dt = 1-cos(t), (dx/dt)² = 1 - 2cos(t) + cos²(t)

dy/dt = sin(t), (dy/dt)² = sin²(t)

(dx/dt)² + (dy/dt)² = 1 - 2cos(t) + cos²(t) + sin²(t) = 2 - 2cos(t) , since cos²(t) + sin²(t) = 1

= 4 sin²(t/2)

S(t) = ∫2sin(t/2)dt from 0 to 2π

= 2(2)(-cos(t/2)) from 0 to 2π = 4 (-(-1) - (-1)) = 4(2) = 8