PGFPlots — a solid of revolution as a 2D-plot

Question

I'm trying to draw something like this:

And I probably should use a 3D-plot as shown here but it seems to be an overkill solution in my case. So I wonder if there is a better way to draw ellipses than mine solution:

\documentclass[12pt, border=0.5mm]{standalone}
\usepackage{graphicx}
\usepackage[ngerman]{babel}
\usepackage[T1]{fontenc}
\usepackage[sc]{mathpazo}    
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

\begin{tikzpicture}[x=1mm, y=1mm]
\begin{axis}[
    axis x line=center,
    axis y line=center,
    xlabel=$x$,
    ylabel=$y$,
    xmin=-0.2,
    xmax=7.5,
    ymin=-1.2,
    ymax=1.2,
    xlabel style={below right},
    ylabel style={above left},
    xticklabel=\empty,
    ytick={-1, 1},  
    height=80mm,
    width=100mm,
]

    \addplot[peter_river, line width=0.25mm, mark=none, samples=150, domain=0:7] {1/x};
    \addplot[peter_river, line width=0.25mm, mark=none, samples=150, domain=0:7] {-1/x};



    \node[font=\small, anchor=east] at (axis cs:7, 1) {$\pi\cdot\displaystyle\int_{1}^{\infty}{\bigg(\displaystyle\frac{1}{x}\bigg)^2}dx$}; 

    \draw[] (axis cs:2, 0) ellipse (20 and 10);

    \draw[] (axis cs:1, 1) -- (axis cs:2, 1);
    \draw[] (axis cs:1, -1) -- (axis cs:2, -1);
    \begin{scope}
        \clip[] (axis cs:1, -1) rectangle (axis cs:1.5, 1);
        \draw[dashed] (axis cs:1, 0) ellipse (20 and 10);
    \end{scope}     
    \begin{scope}
        \clip[] (axis cs:1, -1) rectangle (axis cs:0.5, 1);
        \draw[] (axis cs:1, 0) ellipse (20 and 10);
    \end{scope} 

    \draw[] (axis cs:4, 0) ellipse (10 and 5);

    \draw[] (axis cs:2, 0.5) -- (axis cs:4, 0.5);
    \draw[] (axis cs:2, -0.5) -- (axis cs:4, -0.5);
    \begin{scope}
        \clip[] (axis cs:2, -1) rectangle (axis cs:2.5, 1);         
        \draw[dashed] (axis cs:2, 0) ellipse (10 and 5);
    \end{scope}
    \begin{scope}
        \clip[] (axis cs:2, -1) rectangle (axis cs:1.5, 1);         
        \draw[] (axis cs:2, 0) ellipse (10 and 5);
    \end{scope}

    \draw[] (axis cs:4, 0.25) -- (axis cs:7, 0.25);
    \draw[] (axis cs:4, -0.25) -- (axis cs:7, -0.25);
    \begin{scope}
        \clip[] (axis cs:4, -1) rectangle (axis cs:4.5, 1);         
        \draw[dashed] (axis cs:4, 0) ellipse (5 and 2.5);
    \end{scope}
    \begin{scope}
        \clip[] (axis cs:4, -1) rectangle (axis cs:1.5, 1);         
        \draw[] (axis cs:4, 0) ellipse (5 and 2.5);
    \end{scope}

    \node[font=\small] at (axis cs:1.5, 1.1) {$\pi$};   
    \node[font=\scriptsize] at (axis cs:3, 0.65) {$\displaystyle\frac{\pi}{2}$};    
    \node[font=\scriptsize] at (axis cs:5.5, 0.4) {$\displaystyle\frac{\pi}{4}$};   

    \draw[-stealth] (axis cs:0.5, 0.05) to[out=90, in=-20] (axis cs:0.35, 0.15) 
                        to[out=200, in=90] (axis cs:0.2, 0) 
                        to[out=270, in=160] (axis cs:0.35, -0.2)
                        to[out=20, in=270] (axis cs:0.5, -0.05);
\end{axis}
\end{tikzpicture}

\end{document}

Answer 1

Here's an attempt in Metapost. It's a bit less structured perhaps than PGFplots+Tikz, and maybe this gives you more artistic freedom, for this sort of semi-numerical diagram. I would recommend Asymptote for anything more demanding in 3D than these cyclinders.

In this example, I've used luamplib to make it easier to typeset using Palatino as in the OP, so this should be compiled with lualatex. If you don't use lualatex for your main document, you can either use it to produce a standalone PDF graphic from this example, or adapt the input either to use the gmp package (which works with pdflatex) or run Metapost on it's own. See the linked documentation for details.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\usepackage{unicode-math}
\setmathfont{TeX Gyre Pagella Math}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
numeric u, v; 
u = 3cm;
v = 2cm;

path xx, yy;
xx = (1/4 left -- 19/4 right) scaled u;
yy = (3/2 down -- 3/2 up) scaled v;

drawarrow xx;
drawarrow yy;

draw (left--right) scaled 2 shifted (0,v);
draw (left--right) scaled 2 shifted (0,-v);
label.lft("$1$", (-2,v));
label.lft("$-1$", (-2,-v));

vardef f(expr x) = if x=0: infinity else: (1/x)**2 fi enddef;
path ff;
numeric s;
s = 1/16;
ff = ((12s, f(12s)) for x=13s step s until 73s+eps: -- (x,f(x)) endfor ) 
     xscaled u yscaled v;


path e[]; 
e0 = fullcircle rotated 90 xscaled 24;
e1 = e0 yscaled 2(f(1)*v)   shifted (1u,0);   e2 = e1 shifted (1/2u,0);

e3 = e0 xscaled 1/3;
e4 = e3 yscaled 2(f(3/2)*v) shifted (3/2u,0); e5 = e4 shifted (   u,0);

e6 = e3 xscaled 1/4;
e7 = e6 yscaled 2(f(5/2)*v) shifted (5/2u,0); e8 = e7 shifted (   2u,0);


drawoptions(withpen pencircle scaled .2 withcolor .7 white);
for t= s step  s until 4: draw point t of e1 -- point t of e2; endfor
drawoptions();
draw e2 -- subpath (0,4) of e1 -- point 4 of e2;

drawoptions(withpen pencircle scaled .2 withcolor .7 white);
for t=2s step 2s until 4: draw point t of e4 -- point t of e5; endfor
drawoptions();
draw e5 -- subpath (0,4) of e4 -- point 4 of e5;

drawoptions(withpen pencircle scaled .2 withcolor .7 white);
for t=4s step 4s until 4: draw point t of e7 -- point t of e8; endfor
drawoptions();
draw point 0 of e8 -- subpath (0,4) of e7 -- point 4 of e8;

draw ff withcolor .67 red;
draw ff reflectedabout(left,right) withcolor .67 red;


drawarrow subpath(-3/2,11/2) of e0 xscaled 1/2 yscaled v shifted (1/2u,0);

label.rt ("$x$", point 1 of xx);
label.top("$y$", point 1 of yy);

label(textext("$\displaystyle \pi \int_1^\infty \frac{1}{x^2}\,dx$") scaled 1.2, (4u,5/4v));

label.top("$\pi$",   1/2[point 0 of e1, point 0 of e2]);
label.top("$\pi/2$", 1/2[point 0 of e4, point 0 of e5]);
label.top("$\pi/4$", 1/2[point 0 of e7, point 0 of e8]);

endfig;
\end{mplibcode}
\end{document}