Sombrear regiones entre curvas

Este video está dedicado a explicar cómo construir gráficos que tienen regiones sombreadas, para ilustrar, por ejemplo integrales definidas o regiones que forman un sólido de revolución, todo esto, en LaTeX.

\documentclass{article}
\usepackage{tikz,color}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[
    width=10cm,
    height=6cm,
    xmin=-1,
    xmax=5,
    ymin=0,
    ymax=3.5,
    xlabel=$x$, 
    ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1.05)}},
    ylabel style={at={(ticklabel* cs:1.05)}},
    axis x line=center,
    axis y line=center,]
    \addplot[cyan,thick,samples=200,domain=0:5]{1+sqrt(x)};
    \filldraw[fill=cyan!20,draw=black]
    (0,0) -- plot [domain=0:4] (\x,{1+sqrt(\x)}) --(4,0)-- cycle;
  \end{axis}
\end{tikzpicture}

\begin{tikzpicture}
  \begin{axis}[
    width=10cm,
    height=6cm,
    xmin=-0.5,
    xmax=2.5,
    ymin=-0.3,
    ymax=1.2,
    axis x line=center,
    axis y line=center,
    xlabel=$x$, 
    ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1.05)}},
    ylabel style={at={(ticklabel* cs:1.05)}}]
    \filldraw[draw=black,fill=cyan!20]
    plot [smooth,domain=0:1] (\x,{2*(\x)-(\x)^2})
    -- plot [smooth,domain=1:0] (\x,\x^4)
    -- cycle;
    
    \addplot[cyan,thick,samples=200,domain=-0.2:2]{2*(\x)-(\x)^2};
    \addplot[cyan,thick,samples=200,domain=-0.2:1.1]{(\x)^4};
    
  \end{axis}
\end{tikzpicture}
% region entre 3 curvas

\begin{tikzpicture}
  \begin{axis}[
    width=10cm,
    height=10cm,
    xmin=-0.5,
    xmax=3.5,
    ymin=-1.3,
    ymax=2.2,
    axis x line=center,
    axis y line=center,
    xlabel=$x$, 
    ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1.05)}},
    ylabel style={at={(ticklabel* cs:1.05)}}]
    \addplot[cyan,thick,samples=200,domain=0:1]{(\x)^2};
    \addplot[cyan,thick,samples=200,domain=0:2.1]{(\x)^0.5-1};    
    \addplot[cyan,thick,samples=200,domain=-0.2:3]{-0.5*(\x)+1.5};
    \filldraw[draw=black,fill=cyan!20, opacity=0.5]
    plot [smooth,domain=0:1] (\x,{(\x)^2})-- plot [smooth,domain=1:2.1] (\x,{-0.5*(\x)+1.5})-- plot [smooth,domain=2.1:0] (\x,{(\x)^(1/2)-1})--cycle;
    
  \end{axis}
\end{tikzpicture}
\end{document}