Centering and spacing two subscript lines under a double integral

You can define a general purpose macro \widerof, with three arguments:

1. (optional, default c) the alignment in the box;
2. the symbol to print;
3. the symbol to compare the width with.

In the example I changed 0 to -1 just to show all types of alignment. I also used \smashoperator from mathtools.

\documentclass{amsbook}
\usepackage{mathtools}
\usepackage[frenchstyle]{kpfonts}

\makeatletter
\newcommand{\widerof}[3][c]{\mathpalette\widerof@{{#1}{#2}{#3}}}
\newcommand{\widerof@}[2]{\widerof@@{#1}#2}
\newcommand{\widerof@@}[4]{%
  % #1=math style, #2=alignment, #3=symbol to print, #4=symbol to compare to
  \begingroup
  \sbox\z@{$\m@th#1#3$}%
  \sbox\tw@{$\m@th#1#4$}%
  \makebox[\ifdim\wd\z@>\wd\tw@ \wd\z@\else \wd\tw@\fi][#2]{$\m@th#1#3$}%
  \endgroup
}
\makeatother

\begin{document}
\[
V_n=
V_{n-2}
\smashoperator{\iint\limits_{
  \substack{
    \widerof[r]{0}{-1} < \widerof{\theta}{r} < \widerof[l]{2\pi}{1} \\
    \widerof[r]{-1}{0} < \widerof{r}{\theta} < \widerof[l]{1}{2\pi}
  }
}} (1 - r^2)^{\frac{n-2}{2}} r\D{\theta}\D{r}
\]
\end{document}

Here, I use a TABstack. If you want the lines with less vertical gap, that can easily be done, as an optional argument to \tabularstackunder (default 3pt gap).

\documentclass{amsbook}
\usepackage{tabstackengine}
\usepackage[frenchstyle]{kpfonts}
\usepackage{mathtools}
\TABstackMath
\setstacktabulargap{0pt}
\begin{document}
\[
\TABstackMathstyle{\scriptstyle}
    V_n
    = V_{n-2} \iint\limits_{\mathclap{\tabularstackunder{rcl}{
                                      0<&\theta&<2\pi}{
                                      0<&r&<1}}}
     (1 - r^2)^{\frac{n-2}{2}} r\D{\theta}\D{r}
\]
\end{document}

Use alignedat environment.

\documentclass{amsbook}
\usepackage[frenchstyle]{kpfonts}
\usepackage{mathtools}

\begin{document}
\[
  V_n = V_{n-2} \iint\limits_{\scriptsize\mathclap{
    \begin{alignedat}{2}
      0 &< \theta & &< 2\pi \\[-.6em]
      0 &< \:r & &< 1
    \end{alignedat}
  }}
  (1 - r^2)^{\frac{n-2}{2}} r\D{\theta}\D{r}
\]
\end{document}