Shading area below a curve

Question

As per a suggestion, I'm splitting my previous (multiple) question into separate parts.

I'm trying to create a two graph Figure with pgfplots but the result is not exactly what I want. I want part of the area below a curve to be shaded, but using \closedcycle results in the appearance of some extraneous lines (the vertical line in the gray area and a horizontal line on top of the x-axis which is barely visible). My code is

\documentclass{article}
\usepackage{tikz,pgfplots}
\usepgfplotslibrary{fillbetween}
\pgfplotsset{compat = newest}
\begin{document}
\begin{figure}
   \begin{tikzpicture}
      \begin{axis}[
         ticks=none, 
         axis x line=bottom,
         axis y line=left,
         axis y discontinuity=crunch,
         xmin=0,xmax=1.3,
         ymin=0.5,ymax=1.3]
      \addplot[
         domain = 0:sqrt(1/3),
         samples =200,
         ] {sqrt(7/6-x^2)};
      \addplot[
         domain = sqrt(1/3):sqrt(7/6),
         fill = gray,
         fill opacity = 0.1,
         samples =200,
         y filter/.expression={x==sqrt(7/6 )?0:y},
         ] {sqrt(7/6-x^2)}\closedcycle;
      \addplot[
         domain = 0:1.3,
         samples =200,
         ] {sqrt(5/6)};
      \fill[gray, opacity = 0.1] (0,0) -- (0,0.9129) -- (0.5774,0.9129) -- (0.5774,0);
      \end{axis}
   \end{tikzpicture}
\end{figure}
\end{document}