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\noindent {\bf Problem statement} a) Suppose $f(x,y,z)$ is defined in
a region $\cal R$ of ${\bf R}^3$. The average value of $f$ over $\cal
R$ is ${\int\! \! \int \!\!  \int_{\cal R} f(x,y,z) \, dv}$ divided by
${\int \!\! \int \!\!  \int_{\cal R} 1 \, dv}$.

\smallskip

{\parindent=.3in \narrower \noindent
{\sl The bat flies $\ldots\,$} A bat flies in and around a
hemispherical cave, with water at the bottom, so it cannot land
there. If the radius of the cave is $R$, what is the average height of
the cave to the bat?
(The bat flies totally at random throughout all of the space
available to it.)\par}

\medskip

\noindent b) Suppose $f(x,y,z)$ is defined on a surface $\cal S$ in
${\bf R}^3$. The average value of $f$ over $\cal S$ is
${\int\!\!\int_{\cal S} f(x,y,z) \, dS}$ divided by ${\int \!\!
\int_{\cal S} 1 \, dS}$.

\smallskip

{\parindent=.3in \narrower \noindent
{\sl The slug crawls $\ldots\,$} A {\it non-swimming} slug crawls about
on the inner surface of the same cave as described above. Its motion
is confined to that surface of the cave. What is the average height of
the cave to the slug? (The slug crawls totally at random throughout
all of the space available to it.)\par}

\medskip

\noindent c) Which creature is higher (on average)?





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