\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Suppose $f(x)$ is defined on $0
\le x \le 1$ by the following rule: 

\medskip 

\centerline{$f(x)$ is the first digit in the decimal expansion for
$x$.}

\medskip

\noindent For example, $f(1/2)=5$ and $f(0.719)= 7$.  Sketch the graph
of $y=f(x)$ on the unit interval with appropriate scales for $x$ and
for $y$. Use a graphical interpretation of the definite integral to
compute $\int_0^1 f(x)\, dx$.

\medskip

\noindent c) Suppose the function $g(x)$ is defined as follows:

\medskip

\centerline{$g(x)$ is the second digit in the decimal expansion for
$x$.}

\medskip

\noindent For example, $g(0.437)=3$. Compute $\int_0^1
g(x)\, dx$. Again, a graph may help. 

\vfil\eject\end

The value of $\int_0^1 f(x)g(x)\, dx$ is equal to the product of the
integrals of the factors! Perhaps this is best not requested.