Problem source

In the figure, the large circle with centre $O$ has radius $4$ and
the small circle with centre $P$ has radius $1$. The small circle
rolls around the inside of the larger one. When $P$ was on the line
$OA$ (before the small circle began to roll), the point $B$ was
in contact with the point $A$ on the large circle. 
\begin{center}
\begin{tikzpicture}[scale=0.7]
    % Main circle centered at origin
    \draw (0,0) circle (4);
    
    % Define points
    \coordinate (O) at (0,0);
    \coordinate (A) at (4,0);
    \coordinate (P) at ({3*cos(50)},{3*sin(50)});
    \coordinate (Q) at ({5*cos(50)},{5*sin(50)});
    \coordinate (B) at ({3*cos(50) + 1*cos(170)}, {3*sin(50) + 1*sin(170)});
    
    % Smaller circle centered at P
    \draw (P) circle (1);
    
    % Line from origin to A
    \draw (O) -- (A);
    
    \draw[-{Stealth[length=3mm]}] 
         ($(P) + ({0.7*cos(210)},{0.7*sin(210)})$) 
         arc (210:80:.7);
    % Line from origin to B
    \draw (O) -- (Q);
    
    % Angle phi
    \draw pic["$\phi$", draw=gray!80, angle radius=0.65cm] {angle=A--O--P};
    
    % Labels
    \node[left] at (O) {$O$};
    \node[right] at (A) {$A$};
    \node[right] at (P) {$P$};
    \node[left] at (B) {$B$};
    
    % Points (smaller than before to match reference)
    \fill (O) circle (2.5pt);
    \fill (P) circle (2.5pt);
    \fill (B) circle (2.5pt);
\end{tikzpicture}
\end{center}
Sketch the curve $C$ traced by $B$ as the circle rolls. Show that
if we take $O$ to be the origin of cartesian coordinates and the
line $OA$ to be the $x$-axis (so that $A$ is the point $(4,0)$)
then $B$ is the point 
\[
(3\cos\phi+\cos3\phi,3\sin\phi-\sin3\phi).
\]
It is given that the area of the region enclosed by the curve $C$
is 
\[
\int_{0}^{2\pi}x\frac{\mathrm{d}y}{\mathrm{d}\phi}\,\mathrm{d}\phi,
\]
where $B$ is the point $(x,y).$ Calculate this area.