pgfplots, tikz: How to use fillbetween at a Bezier-curve

Question

With normal curves it works all fine, but what's about Bezier curves?
I have a Bezier curve of the kind (-0.3,3.7) .. controls (2.3,0.9) and ......
I want to set two coordinates on the Bezier-path und fill the area between x-axis and the part of the curve.
What do I have to make better?

\documentclass[margin=3mm, tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usepgfplotslibrary{fillbetween}
%\usepgfplotslibrary{patchplots}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{calc}

\begin{document}
\begin{tikzpicture}[
MyMarks/.style={
decoration={markings,
mark=at position 0.30 with {\coordinate[label=A] (A); \fill[red] circle [radius=2pt];},
mark=at position 0.55 with {\coordinate[label=B] (B); \fill[red] circle [radius=2pt];}
}, postaction={decorate},
}, 
]
\begin{axis}[
axis lines=middle,
xlabel=$x$, ylabel=$y$,
xlabel style = {anchor=north east},
ylabel style = {anchor=north east},
xtick=\empty, ytick=\empty,
clip=false,
xmin=0,xmax=10,
ymin=0, ymax=5,
]
% Curve
\draw[thick, red, name path=Curve,
MyMarks, 
](-0.3,3.7) .. controls (2.3,0.9) and (3.1,3.9) .. (5.1,4) .. controls (6.5,4.1) and (6.5,3) .. (8.2,2) 
node[black, above=15pt,pos=.9]{$f(x)$};
% "x values" of the Points A, B
\coordinate[label=below:Bz] (Bz) at ($(0,0)!(B)!(5,0)$);
\coordinate[label=below:Az] (Az) at ($(0,0)!(A)!(5,0)$);

% fill 1
\draw [cyan, ultra thick, name path=Line1] (A)--(Bz);
\addplot [orange]  fill between [of=Curve and Line1,
soft clip={(Az) rectangle (B)}, % ????
];

% fill 2
\fill[orange!44] (Az) -- (Bz) -- (A) --cycle;

%% Annotations
\draw[dashed]   (A) -- (Az) %node[below] {$a$}
                         (B) -- (Bz) %node[below] {$b$}
                         ;
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

Not an answer but too long for a comment.

---

I am not 100% sure, but I think this is because there are "so less points". PGFPlots "guesses" were the intersection is and when points are far away from each other then the "guess" is worse than when the points are not that far away from each other.

To "prove" that I replicated that with an \addplot having around the same number of points which gives just a little bit better result than yours. If I just add one more point between the two points where the intersection lies, have a look at the result than (by uncommenting the commented line in the table).

% used PGFPlots v1.17
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
    \usetikzlibrary{
        calc,
        decorations.markings,
        patterns.meta,
        %
        pgfplots.fillbetween,
    }
    \pgfplotsset{compat=1.17}
\begin{document}
\begin{tikzpicture}[
    MyMarks/.style={
        decoration={
            markings,
            mark=at position 0.30 with {\coordinate[label=A] (A); \fill[red] circle [radius=2pt];},
            mark=at position 0.55 with {\coordinate[label=B] (B); \fill[red] circle [radius=2pt];}
        },
        postaction={decorate},
    },
]
    \begin{axis}[
        axis lines=middle,
        xlabel=$x$, ylabel=$y$,
        xlabel style = {anchor=north east},
        ylabel style = {anchor=north east},
        xtick=\empty, ytick=\empty,
        clip=false,
        xmin=0,xmax=10,
        ymin=0,ymax=5,
    ]

        \path
            coordinate (X1) at (-0.3,3.7)
            coordinate (X2) at (2.3,0.9)
            coordinate (X3) at (3.1,3.9)
            coordinate (X4) at (5.1,4)
            coordinate (X5) at (6.5,4.1)
            coordinate (X6) at (6.5,3)
            coordinate (X7) at (8.2,2)
        ;
        \pgfplotsinvokeforeach {1,2,3,4,5,6,7}{
            \fill [blue] (X#1) circle [radius=2pt];
        }

        % Curve
        \draw[
            thick, red, name path=Curve, MyMarks,
        ] (X1) .. controls (X2) and (X3) .. (X4)
               .. controls (X5) and (X6) .. (X7)
            node [black, above=15pt,pos=.9]{$f(x)$}
        ;
        % "x values" of the Points A, B
        \coordinate [label=below:Bz] (Bz) at ($(0,0)!(B)!(5,0)$);
        \coordinate [label=below:Az] (Az) at ($(0,0)!(A)!(5,0)$);

        % fill 1
        \draw [cyan, ultra thick, name path=Line1] (A)--(Bz);
        \addplot [orange]  fill between [
            of=Curve and Line1,
            soft clip={(Az) rectangle (B)}, % ????
        ];

        % fill 2
        \fill [orange!44] (Az) -- (Bz) -- (A) --cycle;

        %% Annotations
        \draw [dashed]    (A) -- (Az) %node[below] {$a$}
                          (B) -- (Bz) %node[below] {$b$}
        ;

        % uncomment the commented line and have a look then
        \addplot+ [
            green,
            mark=o,
            smooth,
            name path=Curve2,
        ] table {
            x       y
            -0.3    3.7
            1.8     2.5
%            3.0     3.0
            5.1     4.0
            7.0     3.0
            8.2     2.0
        };

        \addplot [
            pattern={Lines[angle=45]},
            pattern color=blue,
        ] fill between [
            of=Curve2 and Line1,
            soft clip={(Az) rectangle (B)}, % ????
        ];
    \end{axis}
\end{tikzpicture}
\end{document}