Shading a region below two edges in tikz?

Question

Question: In the following figure, how can I shade the region marked "here"?

I'm guessing there is an elegant solution via the intersections library, but I'm not sufficiently familiar with it.

Here's a minimal working example of my LaTeX code:

\documentclass[crop]{standalone}

\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
  \draw[thin,dashed,<->] (0.5,0) to (7.5,0);

  \coordinate (a) at (1,0);
  \coordinate (b) at (3,0);
  \coordinate (c) at (5,0);
  \coordinate (d) at (7,0);

  \draw (a) to[bend left=80] (c);
  \draw (b) to[bend left=80] (d);

  \node (x) at (4,0.3) {\small here};
\end{tikzpicture}

\end{document}

I don't need the word "here", it's just there to point to the region I want shaded.

Answer 1

\begin{tikzpicture}
  \draw[thin,dashed,<->] (0.5,0) to (7.5,0);

  \coordinate (a) at (1,0);
  \coordinate (b) at (3,0);
  \coordinate (c) at (5,0);
  \coordinate (d) at (7,0);

  \draw (a) to[bend left=80] (c);
  \draw (b) to[bend left=80] (d);

  \clip (a) to [bend left=80] (c) -- cycle;
  \fill[yellow]  (b) to[bend left=80] (d) -- cycle;

  \node (x) at (4,0.3) {\small here};
\end{tikzpicture}

Update

In the above solution the right edge of the yellow region is thinner, due to the clipping. It can be solved if the arcs are drawn after the yellow fill. It is neccesary to use a scope to "undo" the clipping when drawing the arcs:

\begin{tikzpicture}
  \draw[thin,dashed,<->] (0.5,0) to (7.5,0);

  \coordinate (a) at (1,0);
  \coordinate (b) at (3,0);
  \coordinate (c) at (5,0);
  \coordinate (d) at (7,0);
  \begin{scope}
    \clip (a) to [bend left=80] (c) -- cycle;
    \fill[yellow]  (b) to[bend left=80] (d) -- cycle;
  \end{scope}
  \draw (a) to[bend left=80] (c);
  \draw (b) to[bend left=80] (d);
  \node (x) at (4,0.3) {\small here};
\end{tikzpicture}

Update 2: Look ma, no clipping!

And of course this is big cheating. Just for fun.

\begin{tikzpicture}
  \draw[thin,dashed,<->] (0.5,0) to (7.5,0);

  \coordinate (a) at (1,0);
  \coordinate (b) at (3,0);
  \coordinate (c) at (5,0);
  \coordinate (d) at (7,0);

  \fill[yellow] 
        (a) to[bend left=80]  (c);
  \fill[white, even odd rule] 
        (a) to[bend left=80] (c) -- (b) to[bend left=80]  (d);
  \draw (a) to[bend left=80] (c);
  \draw (b) to[bend left=80] (d);

  \node (x) at (4,0.3) {\small here};
\end{tikzpicture}

Answer 2

Well it's still cheating but I guess not that bad. You can scope the drawing to keep the clipping local.

\documentclass[crop,tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \draw[thin,dashed,<->] (0.5,0) to (7.5,0);

  \coordinate (a) at (1,0);
  \coordinate (b) at (3,0);
  \coordinate (c) at (5,0);
  \coordinate (d) at (7,0);

  \draw (a) to[bend left=80] (c);
  \clip[preaction={draw}] (b) to[bend left=80] (d);

  \node (x) at (4,0.3) {};
  \shade (a) to[bend left=80] (c);

\end{tikzpicture}
\end{document}