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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 4}\quad {\sl September 23, 1999}

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\begin{enumerate}

\item Let $f$ be a bounded function on $[a,b]$. Completely define the
lower integral of $f$, $\underline{I}(f)$, and the upper integral of $f$
$\overline{I}(f)$.

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\item Let $A$ be the open interval $(0,1) = \{x\in\R\,|\, 0 < x < 1\}$.
Prove that $\sup A = 1$.

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\item Show how to use integrals of step functions to find an approximation
for \[ \int_0^1 \sqrt{x}\,dx \] with an error less than $\pm0.1$. (You do not
need to carry out the arithmetic to get the final number, but you should
include all other steps.)

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