%% 
%%  Der Mathematiksatz mit LaTeX, 1. Auflage 2009
%% 
%%  Example 10-5-2 on page 220.
%% 
%%  Copyright (C) 2009 Herbert Voss
%% 
%%  It may be distributed and/or modified under the conditions
%%  of the LaTeX Project Public License, either version 1.3
%%  of this license or (at your option) any later version.
%% 
%%  See http://www.latex-project.org/lppl.txt for details.
%% 

\documentclass[]{ttctminimal}
\pagestyle{empty}
\setcounter{page}{6}
\setlength\textwidth{355.65944pt}
\AtBeginDocument{\setlength\parindent{0pt}}
\StartShownPreambleCommands
\usepackage{amsmath,esint} \newcommand*\diff{\mathop{}\!\mathrm{d}}
\StopShownPreambleCommands
\begin{document}
\begin{align*}
\biggl(\int_{-\infty}^\infty e^{-x^2}\,dx\biggr)^2
  &= \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\diff x\diff y \\
  &= \int_0^{2\pi}\int_0^\infty e^{-r^2}r\diff r\diff\theta                  \\
  &= \int_0^{2\pi}\biggl(-{e^{-r^2}\over2}\bigg\vert_{r=0}^{r=\infty}\,\biggr)\diff\theta\\
  &= \pi                                          \tag*{q.e.d.}
\end{align*}
\end{document}