Making Half a ellipse dashed - Coloring and area of a cylinder

Question

I'm having some problems drawing ellipse in tikz. Can someone help(show) me, how I can make a graph like this?

I hope someone. could lead me on the way.

1. I can not find out to make some area a color, like between "a" and "b".
2. And the cylinder between "x_0" and "x" with the Oblique lines.
3. I can not find out to make half of the ellipse dashed like around "a"

My code so far:

\documentclass[a4paper,10pt,twoside]{extarticle}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{tikz}
\usepackage{pgf}
\usepackage{amsmath,amssymb}
\usetikzlibrary{arrows,tikzmark}
\usetikzlibrary{calc,trees,positioning,arrows,chains,shapes.geometric,%
    decorations.pathreplacing,decorations.pathmorphing,shapes,%
    matrix,shapes.symbols}
\begin{document}
\begin{center}
\begin{minipage}[t]{0,45\textwidth}\centering
    %\vspace{1cm}
    \definecolor{cqcqcq}{rgb}{0.55,0.55,0.9}
    \definecolor{gitter}{rgb}{0.65,0.65,0.65}
    \centering
    \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm, scale=1] 
        %\draw [color=gitter, xstep=0.5cm,ystep=0.5cm] (-5,-1) grid (5,10);  % Grid område
        \draw[->,color=black, line width=1pt] (-5,0) -- (15,0);    %X-akse
        %\draw[color=black] (-2,0) node[below] {\fontsize{7}{7}$\textbf{-2}$};  % -2 på xaksen
        \draw[color=black] (10,-0.4) node[below] {\fontsize{12}{12}$\textbf{b}$};  % b på xaksen
        \draw[color=black] (-3,-0.4) node[below] {\fontsize{12}{12}$\textbf{a}$};  % a på xaksen
        \draw[color=black] (15,0) node [anchor=west] {\fontsize{7}{7} \text{(1)}}; %Aksenanvn
        \draw[->,color=black, line width=1pt] (0,-10) -- (0,10);    %Y-akse
        \draw[color=black] (0,11.3) node [anchor=north] {\fontsize{7}{7} \text{(2)}}; % Aksenavn
        \clip(-5,-10) rectangle (15,10);
        \draw[smooth,samples=100,domain=-5:15, line width=1pt, color=blue] plot(\x,{0.002*(\x)^3-0.02*(\x)+0.25*(\x)+4});
        \draw[smooth,samples=100,domain=-3:10, line width=1pt, color=blue] plot(\x,{-0.002*(\x)^3+0.02*(\x)-0.25*(\x)-4});
        \draw[color=red,line width=1pt] (10,0) ellipse (0.5cm and 4.1cm); %grænse b
        \draw[color=red,line width=1pt] (-3,0) ellipse (0.5cm and 1.6cm); %Grænse a
        \draw (6,0) ellipse (0.5cm and 2.9cm);
        \draw (2,0) ellipse (0.5cm and 2.9cm);
        \draw (2,5.812) -- (6,5.812);
        \draw (2,-5.812) -- (6,-5.812);
        \draw[-,color=black,dashed, line width=1pt] (6,5.812) -- (6,0);
    \end{tikzpicture}
\end{minipage}

\end{center}

\end{document}

Answer 1

I would recommend not setting the font size explicitly, but use e.g. \large or variants. -not setting line width explicitly, but use thick or variants. -not using color=black as black is default color. ellipse always draws a full ellipse to draw a part, you can use arc. To \fill an area, it needs to be a fully closed curve. -first make a closed curve with \draw and then replace \draw with \fill. Here is a start:

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}
   \draw[->] (-2,0) -- (4,0) node[right] {(1)};
   \draw[->] (0,-3) -- (0,3) node[above] {(2)};
   \draw[color=red!80!black] (-1,0.4) arc [x radius=0.2, y radius=0.4, start angle=90, end angle=270];
   \draw[color=red!80!black, densely dotted] (-1,0.4) arc [x radius=0.2, y radius=0.4, start angle=90, end angle=-90];
   \draw (-1,-0.2) node {a};
   \draw[color=red!80!black] (3,0) ellipse [x radius=0.2, y radius=1];
   \draw (3,-0.2) node {b};
   \draw[blue, thick] (-1.4,0.3) to[out=10, in=190] (-1,0.4) to[out=10, in=210] (3,1) to[out=30, in=240] (4,1.5);
   \fill[red!50!yellow, opacity=0.5] (-1,-0.4) arc [x radius=0.2, y radius=0.4, start angle=-90, end angle=-270] to[out=10, in=210] (3,1) arc [x radius=0.2, y radius=1, start angle=90, end angle=-90] to[out=150, in=-10] cycle;
\end{tikzpicture}
\end{document}