How to find t interval in calculating volume of parametric equations rotated

Question

How to find interval $t$ in calculating volume of parametric equations rotated. $$x=2(t-\sin t),~y=2(1-\cos t)$$ Find the volume as curves are rotated around $x$-axis. The interval of $t$ is not given, Is there any way to calculate $t$?

Answer 1

Since the curve repeats itself at each period $2\pi$ we can take the first period interval for $y(t)$ that is $t\in [0,2\pi]$.

Answer 2

The curve $$x=2(t-\sin t),~y=2(1-\cos t)$$

has its y-intercepts at $$y=2(1-\cos t)=0$$ which includes $t=0$ and $t=2\pi$

Thus if you are interested in the first region rotated, your limits are $t=0$ and $t=2\pi$