Fill Between on Polar Plot

Question

I am trying to recreate a graph from the 2024 AP Daily Practice Session. I need to fill in the area of R and directly below it. I have tried FillBetween Command and Soft Clip, but I am just not able to get the correct Clip. From the graph, I have the R shaded, but you can see there is extra shading above and below. Any help is much appreciated.

Nick

\documentclass[12pt,a4paper]{article}
    \usepackage{amsmath,amsfonts,amssymb}
    \usepackage{parskip}
    \usepackage{fancyhdr}
    \usepackage[margin=1in]{geometry}
    \usepackage{tabularray}
    \usepackage{enumitem}
    \usepackage{soul}
    \usepackage{multicol}
    \usepackage[many]{tcolorbox}
    \usepackage{ragged2e}
    \usepackage{tikz,pgfplots}
    \usetikzlibrary{arrows.meta}
    \usetikzlibrary{backgrounds}
    \pgfplotsset{compat=newest}
    \usetikzlibrary{calc}
    \usepgfplotslibrary{fillbetween}

    
        \pagestyle{fancy}
            \fancyhead[L]{Polar Curves on Calc BC}
            \fancyhead[R]{Name:\enspace\makebox[2in]{\hrulefill}}
            \fancyfoot[C]{BC Calculus}
            \fancyfoot[R]{\thepage}
            \fancyfoot[L]{Created By: Mr. Bennett}
                \renewcommand{\headrulewidth}{0.4pt}
                \renewcommand{\footrulewidth}{2pt}
                \newcommand*{\myprime}{^{\prime}\mkern-1.2mu}
                \newcommand*{\dprime}{^{\prime\prime}\mkern-1.2mu}
                \newcommand*{\trprime}{^{\prime\prime\prime}\mkern-1.2mu}   
        
        \addtolength{\topmargin}{-2.49998pt}
        
%For Polar      
\pgfplotsset{
    interpret as polar/.style={
            x filter/.code=\pgfmathparse{cos(rawx)*rawy},
            y filter/.code=\pgfmathparse{sin(rawx)*rawy}
        }
}

\begin{document}

    \subsection*{\hl{What Do We Need To Know?}}

        \begin{Center}
            \begin{tblr}{width=\textwidth,hlines={1pt},vlines={1pt},colspec={Q[m]Q[m]Q[m]Q[m]}}
            {Conversion \\ Formulas} &  \(x=r\cos\big(\theta\big)\)\, \(y=r\sin\big(\theta\big)\)   &   {Slope of the\\ Tangent to a Polar Curve}   & \(\dfrac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}\)\\
            {Area Bounded \\ by a Polar Curve}  &   \(A=\displaystyle\frac{1}{2}\int_{a}^{b}\left[f\big(\theta\big)\right]^2\, d\theta\)    &   {Distance to the \\ Origin}     & \(r=f\big(\theta\big)\) \\
            \end{tblr}
        \end{Center}    

\begin{Center}
\begin{tikzpicture}
\begin{axis}[
            axis lines=center,
            axis equal image,
            enlargelimits=true
        ]
    \addplot[name path=A,
            red, thick,
            domain=0:360,
            samples=360,
            smooth,
            interpret as polar
        ] 
        (x,{4-3*cos(x)});
    \addplot[name path=B,
            blue, thick,
            domain=0:360,
            samples=360,
            smooth,
            interpret as polar
        ] (x,{5*cos(x)});
        \addplot[fill=blue!30,opacity=0.5]fill between[of = A and B,soft clip={domain=-pi/2:pi/2}];
       \node at (.75,1){\(R\)};
\end{axis}
\end{tikzpicture}
\end{Center}

    \begin{enumerate}
        \item Let \(R\) be the region that is inside the graph of \(r=4-\cos \theta\) and also inside the graph of \(r=5\cos\theta\), as shaded in the figure above.  What is the area of \(R\)? 
            \begin{enumerate}[itemsep=.15in,label=(\Alph*)]
                \item 2.261
                \item 3.764
                \item 11.398
                \item 23.999
            \end{enumerate}
    \end{enumerate}
        \rule{\textwidth}{1pt}
\end{document}

Answer 1

Polar or not, for more complex shapes, I suggest that you use a regular \fill with intersection segments and sequence as follows (place on background layer if needed):

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{calc, backgrounds}
\usepgfplotslibrary{fillbetween}
        
%For Polar      
\pgfplotsset{
    interpret as polar/.style={
        x filter/.code=\pgfmathparse{cos(rawx)*rawy},
        y filter/.code=\pgfmathparse{sin(rawx)*rawy}
    }
}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
            axis lines=center,
            axis equal image,
            enlargelimits=true
        ]
    \addplot[name path=A,
            red, thick,
            domain=0:360,
            samples=360,
            smooth,
            interpret as polar
        ] 
        (x,{4-3*cos(x)});
    \addplot[name path=B,
            blue, thick,
            domain=0:360,
            samples=360,
            smooth,
            interpret as polar
        ] (x,{5*cos(x)});
    \begin{scope}[on background layer]
        \fill[blue!30, opacity=0.5, intersection segments={
                of=A and B, sequence={L1 -- R2 -- L3}
            }] -- cycle;
    \end{scope}
    \node at (.75,1) {\(R\)};
\end{axis}
\end{tikzpicture}
\end{document}

---

To better understand what is meant by L1 -- R2 -- L3, you can check out the following code:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{calc, backgrounds}
\usepgfplotslibrary{fillbetween}
        
%For Polar      
\pgfplotsset{
    interpret as polar/.style={
        x filter/.code=\pgfmathparse{cos(rawx)*rawy},
        y filter/.code=\pgfmathparse{sin(rawx)*rawy}
    }
}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
            axis lines=center,
            axis equal image,
            enlargelimits=true
        ]
    \addplot[name path=A,
            red, thick,
            domain=0:360,
            samples=100,
            smooth,
            interpret as polar
        ] 
        (x,{4-3*cos(x)});
    \addplot[name path=B,
            blue, thick,
            domain=0:360,
            samples=100,
            smooth,
            interpret as polar
        ] (x,{5*cos(x)});
    \begin{scope}[on background layer]
        \fill[blue!30, opacity=0.5, intersection segments={
                of=A and B, sequence={L1 -- R2 -- L3}
            }] -- cycle;

        \draw[->, red, thick, intersection segments={
                of=A and B, sequence={L1}
            }] node[below left] {L1};
        \draw[->, red, thick, intersection segments={
                of=A and B, sequence={L2}
            }] node[below left] {L2};
        \draw[->, red, thick, intersection segments={
                of=A and B, sequence={L3}
            }] node[below left] {L3};

        \draw[->, blue, thick, intersection segments={
                of=A and B, sequence={R1}
            }] node[below right] {R1};
        \draw[->, blue, thick, intersection segments={
                of=A and B, sequence={R2}
            }] node[below right] {R2};
        \draw[->, blue, thick, intersection segments={
                of=A and B, sequence={R3}
            }] node[below right] {R3};
    \end{scope}
    \node at (.75,1) {\(R\)};
\end{axis}
\end{tikzpicture}
\end{document}

You can see that L1 means "first segment of the left path (which is A here)" where segments means the part between the intersections of the paths starting from the rightmost point on the x axis and going counterclockwise.