Find the volume under the parametric surface

Question

I have a surface defined in $(x,y,z)$ by $$(a(t-\sin(t)),a(1-\cos(t))\cos(\theta),a(1-\cos(t))\sin(\theta))$$ Where both $\theta$ and $ t$ varies between $[0,2\pi]$ how can I find the volume bounded by this surface? I started from the equations $$x(t)=a(t-\sin(t))\qquad y(t)=a(1-\cos(t)).$$

Answer 1

The surface is obtained by revolving the plane curve $$(x(t),y(t))=a(t-\sin(t), 1−\cos(t))\quad t\in [0,2\pi],$$ which is a Cycloid in parametric form, around the $x$-axis.

In order to compute the volume see HERE: $$V=\pi\int_0^{2\pi} y(t)^2 x'(t)dt=a^3\pi\int_0^{2\pi} (1−\cos(t))^2 \cdot(1-\cos(t)) dt.$$ Can you take it from here?