id: 05845503
dt: j
an: 2013a.00597
au: Pippenger, Nicholas
ti: Two extensions of results of Archimedes.
so: Am. Math. Mon. 118, No. 1, 66-71 (2011).
py: 2011
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: G40
ut: method of Archimedes; Cavalieri’s principle; intersection of cylinders;
hoof
ci:
li: doi:10.4169/amer.math.monthly.118.01.066
ab: Let us consider the following bodies: { indent=4mm \item{‒} $X$ is a
filled right circular cylinder of radius 1 about the $x$-axis.
\item{‒} $Y$ is a similar cylinder about the $y$-axis. \item{‒} $Z$
is a similar cylinder about the $z$-axis. \item{‒} $R=X\cap Y$ and
$S=X\cap Y\cap Z$. \item{‒} $U$ is the filled right isosceles
triangular cylinder with side 1 given by the inequalities $z\geq 0$,
$z\leq y$ and $y\leq 1$. \item{‒} $H=U\cap Z$ (the so-called
“hoof”). \item{‒} $V$ is the filled right isosceles triangular
cylinder with side 1 given by the inequalities $z\geq 0$, $z\leq x$ and
$x\leq 1$. \item{‒} $J=U\cap V\cap Z$. } In this paper the author
uses Cavalieri’s principle to compute rigorously the following
volumes: { indent=6mm \item {(1)} vol$(R)=16/3$. \item {(2)}
vol$(S)=16-8\sqrt{2}$. \item {(3)} vol$(H)=2/3$. \item {(4)}
vol$(J)=(2-\sqrt{2})/3$. } Bodies $R$ and $H$ both appear in
Archimedes’ method where, obviously, different techniques are used to
compute their volume. The volume of $S$ and $H$ are presented here as
generalizations of the previous ones using Cavalieri’s principle.
rv: Antonio M. Oller (Zaragoza)