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\documentclass{standalone}
\usepackage{tikz}
\usepackage{pgfplots,pgfplotstable}
\pgfplotsset{compat=newest}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
	\begin{axis}[
		domain=-2:2,
		samples=500,
		xmin=-2.2, xmax=2.2, xtick={1},
		ymin=-.1, ymax=1.1, ytick=\empty, yscale=.5,
		axis y line=center,
        axis x line=middle,
		]
		\addplot[draw=none,fill=yellow!30,domain=0:1]{exp(-1*x^2)}\closedcycle;
		\addplot+[name path=exp, mark=none,color=black] {exp(-1*x^2)};
		\path[name path=eins] (1,0)--(1,1);
		\path[name intersections={of= eins and exp, by=schnitt}];
		\draw[dashed] (1,0)--(schnitt);
	\end{axis}
\end{tikzpicture}
\end{document}

\documentclass{standalone} 
\usepackage{pgfplots}% lädt tikz, ...
\pgfplotsset{compat=newest} 
\begin{document} 
\begin{tikzpicture} 
   \begin{axis}[
      axis on top,% <- ergänzt
      domain=-2:2, 
      samples=500, 
      xmin=-2.2, xmax=2.2, xtick={1}, 
      ymin=-.1, ymax=1.1, ytick=\empty, yscale=.5, 
      axis y line=center, 
        axis x line=middle, 
      ] 
      \addplot[draw=none,fill=yellow!30,domain=0:1]{exp(-1*x^2)}\closedcycle; 
      \addplot+[mark=none,color=black] {exp(-1*x^2)}; 
      \draw[dashed] (1,0)--(1,{exp(-1)}); 
   \end{axis} 
\end{tikzpicture} 
\end{document}