%% Created by Maple 9.5, Windows XP
%% Source Worksheet: z:\tex\UeA\Maß- und Integrationstheorie\Prinzip des Cavalieri\_001\V_ndim_Kugel.mws
%% Generated: Tue Dec 18 15:46:29 CET 2007
\documentclass{article}
\usepackage{maplestd2e}
\def\emptyline{\vspace{12pt}}
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\DefineCharStyle{Maple 2D Input}
\DefineCharStyle{Maple Maple Input}
\DefineCharStyle{Maple 2D Output}
\DefineCharStyle{Maple 2D Math}
\DefineCharStyle{Maple Hyperlink}
\begin{document}
\pagestyle{empty}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{V:=proc(n,r)}}}\QTR{Maple Maple Input}{{\normalsize{
 if n=1 then 2*r else 2*r*int(V(n-1,r*sin(phi))*sin(phi), phi=0..Pi/2) fi}}}\QTR{Maple Maple Input}{{\normalsize{
end;}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{V := proc (n, r) if n = 1 then 2*r else 2*r*int(V(n-1, r*sin(phi))*sin(phi), phi = 0 .. 1/2*Pi) end if end proc}{%
\[
V\, := \,\textbf{proc} (nr) \\
\mapleIndent{1} \textbf{if} \,n \, = \, 1 \, \textbf{then} \\
\mapleIndent{2} 2 \ast r\\
\mapleIndent{1} \textbf{else}\\
\mapleIndent{2} 2 \ast r \ast \mathit{int} (\mathit{V} (n-1,\,r \ast \mathit{sin} (phi)) \ast \mathit{sin} (phi),\,phi \, = \, 0..1/2 \ast Pi)\\
\mapleIndent{1} \textbf{end\ if}; \\
\textbf{end\ proc};
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{vol:=[seq(V(n,r),n=1..10)];}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{vol := [2*r, r^2*Pi, 4/3*r^3*Pi, 1/2*r^4*Pi^2, 8/15*r^5*Pi^2, 1/6*r^6*Pi^3, 16/105*r^7*Pi^3, 1/24*r^8*Pi^4, 32/945*r^9*Pi^4, 1/120*r^10*Pi^5]}{%
${\it vol}\, := \,[2\,r,{r}^{2}\pi ,4/3\,{r}^{3}\pi ,1/2\,{r}^{4}{\pi }^{2},{\frac {8}{15}}\,{r}^{5}{\pi }^{2},1/6\,{r}^{6}\\
\mbox{}{\pi }^{3},{\frac {16}{105}}\,{r}^{7}{\pi }^{3},1/24\,{r}^{8}{\pi }^{4},{\frac {32}{945}}\,{r}^{9}\\
\mbox{}{\pi }^{4},{\frac {1}{120}}\,{r}^{10}{\pi }^{5}]$}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{seq(vol[n]/vol[n-1],n=2..10);}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{1/2*r*Pi, 4/3*r, 3/8*r*Pi, 16/15*r, 5/16*r*Pi, 32/35*r, 35/128*r*Pi, 256/315*r, 63/256*r*Pi}{%
\[
1/2\,r\pi ,\,4/3\,r,\,3/8\,r\pi ,\,{\frac {16}{15}}\,r,\,{\frac {5}{16}}\,r\pi ,\,{\frac {32}{35}}\,r,\,{\frac {35}{128}}\,r\pi ,\,{\frac {256}{315}}\,r,\,{\frac {63}{256}}\,r\pi 
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{seq(vol[n]/vol[n-2],n=3..10);}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{2/3*r^2*Pi, 1/2*r^2*Pi, 2/5*r^2*Pi, 1/3*r^2*Pi, 2/7*r^2*Pi, 1/4*r^2*Pi, 2/9*r^2*Pi, 1/5*r^2*Pi}{%
\[
2/3\,{r}^{2}\pi ,\,1/2\,{r}^{2}\pi ,\,2/5\,{r}^{2}\pi ,\,1/3\,{r}^{2}\pi ,\,2/7\,{r}^{2}\pi ,\,1/4\,{r}^{2}\pi ,\,2/9\,{r}^{2}\pi ,\,1/5\,{r}^{2}\pi 
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{S:=proc(n,r)}}}\QTR{Maple Maple Input}{{\normalsize{
 if n=2 then 2*r*Pi else 2*r*int(S(n-1,r*sin(phi)), phi=0..Pi/2) fi}}}\QTR{Maple Maple Input}{{\normalsize{
end;}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{S := proc (n, r) if n = 2 then 2*Pi*r else 2*r*int(S(n-1, r*sin(phi)), phi = 0 .. 1/2*Pi) end if end proc}{%
\[
S\, := \,\textbf{proc} (nr) \\
\mapleIndent{1} \textbf{if} \,n \, = \, 2 \, \textbf{then} \\
\mapleIndent{2} 2 \ast Pi \ast r\\
\mapleIndent{1} \textbf{else}\\
\mapleIndent{2} 2 \ast r \ast \mathit{int} (\mathit{S} (n-1,\,r \ast \mathit{sin} (phi)),\,phi \, = \, 0..1/2 \ast Pi)\\
\mapleIndent{1} \textbf{end\ if}; \\
\textbf{end\ proc};
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{ofl:=[seq(S(n,r),n=2..11)];}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{ofl := [2*r*Pi, 4*r^2*Pi, 2*r^3*Pi^2, 8/3*r^4*Pi^2, r^5*Pi^3, 16/15*r^6*Pi^3, 1/3*r^7*Pi^4, 32/105*r^8*Pi^4, 1/12*r^9*Pi^5, 64/945*r^10*Pi^5]}{%
${\it ofl}\, := \,[2\,r\pi ,4\,{r}^{2}\pi ,2\,{r}^{3}{\pi }^{2},8/3\,{r}^{4}{\pi }^{2},{r}^{5}{\pi }^{3},{\frac {16}{15}}\,{r}^{6}{\pi }^{3},1/3\\
\mbox{}\,{r}^{7}{\pi }^{4},{\frac {32}{105}}\,{r}^{8}{\pi }^{4},1/12\,{r}^{9}{\pi }^{5},{\frac {64}{945}}\\
\mbox{}\,{r}^{10}{\pi }^{5}]$}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{seq(ofl[n]/ofl[n-1],n=2..10);}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{2*r, 1/2*r*Pi, 4/3*r, 3/8*r*Pi, 16/15*r, 5/16*r*Pi, 32/35*r, 35/128*r*Pi, 256/315*r}{%
\[
2\,r,\,1/2\,r\pi ,\,4/3\,r,\,3/8\,r\pi ,\,{\frac {16}{15}}\,r,\,{\frac {5}{16}}\,r\pi ,\,{\frac {32}{35}}\,r,\,{\frac {35}{128}}\,r\pi ,\,{\frac {256}{315}}\,r
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{seq(ofl[n]/ofl[n-2],n=3..10);}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{r^2*Pi, 2/3*r^2*Pi, 1/2*r^2*Pi, 2/5*r^2*Pi, 1/3*r^2*Pi, 2/7*r^2*Pi, 1/4*r^2*Pi, 2/9*r^2*Pi}{%
\[
{r}^{2}\pi ,\,2/3\,{r}^{2}\pi ,\,1/2\,{r}^{2}\pi ,\,2/5\,{r}^{2}\pi ,\,1/3\,{r}^{2}\pi ,\,2/7\,{r}^{2}\pi ,\,1/4\,{r}^{2}\pi ,\,2/9\,{r}^{2}\pi 
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{{\normalsize{seq(ofl[n]/vol[n+1],n=2..9);}}}\end{Maple Normal}}{}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\QTR{Maple 2D Output}{\mapleinline{inert}{2d}{3/r, 4/r, 5/r, 6/r, 7/r, 8/r, 9/r, 10/r}{%
\[
3\,{r}^{-1},\,4\,{r}^{-1},\,5\,{r}^{-1},\,6\,{r}^{-1},\,7\,{r}^{-1},\,8\,{r}^{-1},\,9\,{r}^{-1},\,10\,{r}^{-1}
\]
}
}

\end{maplelatex}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}\QTR{Maple Maple Input}{}\QTR{Maple Maple Input}{}\end{Maple Normal}}{}
\end{mapleinput}

\end{document}
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