I have drawn two shapes in the code below, a red curve, and a black rectangle. I have labeled four points, A, B, C and D. I would like to know how to draw a path along the black rectangle from A to B, along the red curve from B to C, and then back along the black rectangle from C to D back to A. I should mention that I would ultimately like to fill the region enclosed by this shape ABCD.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\draw[name path=p0] (0,.8) node[above left] {$D$} -- ++(3,0) |- (0,2) node[above left] {$A$} -- cycle;
\draw[red,name path=p2] (.2,3) .. controls (.5,3) and (3,.2) .. (3.5,.2);
\fill[name intersections={of=p0 and p2}] (intersection-1) circle (2pt) node[below left] {$C$} (intersection-2) circle (2pt) node[above left] {$B$};
\end{tikzpicture}
\end{document}
