How can I fill below of intersection of a plane and a cylinder of this picture?

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\newcommand{\alp}{60}
\newcommand{\bet}{30}
\fill[gray!20,opacity=.5] ({.5*cos(270+\bet)},{sin(270+\bet)}) -- ({.5*cos(270-\alp)},{sin(270-\alp)}) arc ({270-\alp}:{270+\bet}:.5 and 1);
\fill[gray!15] ({.5*cos(270-\alp)},{sin(270-\alp)}) --             ({2+.5*cos(270-\alp)},{sin(270-\alp)}) 
--({2+.5*cos(270+\bet)},{sin(270+\bet)}) 
--({.5*cos(270+\bet)},{sin(270+\bet)}) 
 --  cycle;
    \fill[gray!20] (
{.5*cos(270+\bet)},
{sin(270+\bet)}
) -- 
(
{2+.5*cos(270+\bet)},
{sin(270+\bet)}
)
arc ({270+\bet}:270:.5 and 1) -- (0,-1) -- cycle;
\draw[gray] (0,0) ellipse (.5 and 1);
\draw[gray] (2,-1) arc (-90:90:.5 and 1);
\draw[gray!20] (2,-1) arc (-90:-270:.5 and 1);
\draw[gray] (0,1) -- (2,1);
\draw[gray] (0,-1) -- (2,-1);
\end{tikzpicture}
\end{document}