Arc length of parametric curves

Problem 8.91 · medium

1. \[ \left[\begin{matrix}\frac{d}{d t} \left(3 t - 3 \sin{\left(t \right)}\right)\\\frac{d}{d t} \left(3 - 3 \cos{\left(t \right)}\right)\end{matrix}\right] = \left[\begin{matrix}3 - 3 \cos{\left(t \right)}\\3 \sin{\left(t \right)}\end{matrix}\right] \]
   Velocity components.✓ Proved
2. \[ \left(3 - 3 \cos{\left(t \right)}\right)^{2} + 9 \sin^{2}{\left(t \right)} = 36 \sin^{2}{\left(\frac{t}{2} \right)} \]
   (dx/dt)² + (dy/dt)², simplified with 1 − cos t = 2 sin²(t/2).✓ Proved
3. \[ \int\limits_{0}^{2 \pi} 6 \sin{\left(\frac{t}{2} \right)}\, dt = 24 \]
   Integrate the speed (sin(t/2) ≥ 0 on [0, 2π]).✓ Proved