%Volume of a ball in R^4 %If you modify this file, please indicate here and in the footnote %Math263D \documentclass[11pt]{article} \usepackage{amsmath} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large The Volume of a Ball in $4$-Space \footnote{Copyright \copyright 2009 Steve Chapin and Todd Young. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} Let $B(a)$ be the (closed) ball centered at the origin of radius $a$ in $\mathbf{R}^{4}$. So, \[ B(a)=\left\{(x,y,z,w)\in \mathbf{R}^{4} : x^{2}+y^{2}+z^{2}+w^{2}\le a^{2}\right\}. \] Let $S(a)$ be the sphere centered at the origin of radius $a$ in $\mathbf{R}^{4}$. So, \[ S(a)=\left\{(x,y,z,w)\in \mathbf{R}^{4} : x^{2}+y^{2}+z^{2}+w^{2}= a^{2}\right\}. \] Let $V(a)$ be the volume of $B(a)$ and $A(a)$ be the (surface) area of $S(a).$ Now, \[ V(a)=\iiiint\limits_{B(a)} dV = \int_{-a}^{a}\!\!\int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} \!\!\int_{-\sqrt{a^{2}-x^{2}-y^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}}} \!\!\int_{-\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}} \,dw\,dz\,dy\,dx. \] Integrating once and using the symmetry we see the iterated integral is equal to \[ 16\int_{0}^{a}\!\!\int_{0}^{\sqrt{a^{2}-x^{2}}} \!\!\int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}}} \,\sqrt{a^{2}-x^{2}-y^{2}-z^{2}} \,dz\,dy\,dx. \] We can use \textsc{Matlab} to compute this iterated integral. \begin{enumerate} \item Enter the following commands:\\ \verb& syms x y z&\\ \verb& syms a positive& \dotfill Makes $a>0$. \\ \verb& 16*int(int(int(sqrt(a^2-x^2-y^2-z^2),z,0,sqrt(a^2-x^2-y^2)), ... & \\ \verb& y,0,sqrt(a^2-x^2)),0,a)& \item Given that \[ V(a)=\int_{0}^{a}A(r)\,dr \] find $A(a)$. Hint: Use the Fundamental Theorem of Calculus. \item Modify the commands above to find the volume of the ball of radius $a$ in $\mathbf{R}^{5}$. Then find the (surface) area of the sphere of radius $a$ in $\mathbf{R}^{5}$. \item Using the formulas that you already know in $\mathbf{R}^{2}$ and in $\mathbf{R}^{3}$, try to come up with a general formula for the volume of the ball of radius $a$ and the (surface) area of the sphere of radius $a$ in $\mathbf{R}^{n}$. (Hint: You will need different formulas for $n$ even and $n$ odd.) \end{enumerate} \vfill \noindent \textsf{The student uses \textsc{Matlab} to assist in computing the volume of balls and the (surface) area of spheres in $\mathbf{R}^{4}$ and $\mathbf{R}^{5}$.} \end{document}