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\centerline{MATH 136 -- Calculus 2}
\centerline{Introducing the Integral -- Total Accumulation and Area Problems}
\centerline{August 31, 2016}
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\noindent
{\it Background}
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Our course will study the second part of the subject of calculus: \emph{integral calculus}.
The first part, differential calculus, starts with a function and studies the instantaneous
rate of change (the derivative).  This part of calculus turns the tables around, so to speak, 
and asks questions like the first one below, where we know the rate at which some quantity is
changing, but we want to know the total accumulation or amount of that quantity over
some period of time.  As we will see, that is
connected with the problem of finding the \emph{area under a graph}.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[(1)] A heavy rainstorm in Portland, Maine in October 1996 resulted in record rainfall\footnote{That 
is, a record for Portland.  It was nothing like as heavy as the rain totals from 
recent storms in Louisiana earlier in August 2016 that caused the devastating flooding that's
still plaguing that part of the country. I wasn't able to find the same sort of data 
for those storms in this form, although there were apparently times in Baton Rouge on August 12 where the rainfall 
rate was something like 7.5 cm/hour(!) Many places in Louisiana had over 70 cm of rain over a
24-hour period -- that's nearly 28 inches(!)} 
The estimated rainfall rate, $R(t)$, in units of cm of rain per hour during the storm is 
given in the following table, where the $t$ values are given in hours after midnight on October 21:
\[
\begin{array}{cccccccc}
t\ {\rm hours} & \vline & 0-2 & 2-4 & 4-9 & 9-12 & 12-20 & 20-24 \\ \hline
R(t)\ {\rm cm/hr} & \vline & 0.5 & 0.3 & 1.0 & 2.5 & 1.5 & 0.6
\end{array}
\]
\begin{enumerate}
\item[(a)]  Estimate the total rainfall in Portland during this 24-hour period.
\item[(b)]  Draw a graph that illustrates the computation you were doing, and that
shows the total accumulation of rain can be visualized as an area.
\end{enumerate}
\item[(2)]  What are some of the basic shapes for which you know area formulas from
high school mathematics?  
\item[(3)]  The rainfall rate in the situation in question (1) was certainly not
constant over each of the time periods in the table -- it was certainly changing 
continuously.  If we had a really accurate minute-by-minute, or second-by-second
reading of the rainfall rate, how could we estimate the total rainfall more closely?
\end{enumerate}
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