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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Discussion -- Total Accumulation}
\centerline{November 10, 2010}
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{\it Background}
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Today, we want to begin to consider the problem of computing the 
``total accumulation'' of a function of several variables.  We
will start by reconsidering the data giving the snow depth
at various locations after the ``Blizzard of 1978'' that we
saw before in the context of density plots.  Here is the 
density plot, and the snow depth data that was used to generate 
it.
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Note: In the following, we will consider a simplified version of the actual
geometry, in which we {\it ignore the curvature of the Earth.}  That is, we
will treat this region as though it was planar.
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\noindent
{\it Discussion Questions}
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\item{A)} Suppose that the snow in a region (again, planar) of
area $A$ square feet is constant of depth = $d$ feet.  What is the total 
volume of snow that fell on that region?  (Use units of cubic feet.)
\item{B)}  If the snow depth is not constant (which of course is 
always the case), but you have readings of snow depth at several 
different locations in the region, how could you {\it approximate}
the total volume of snow that fell on the region?  Would the approximation
involve any choices on your part?  If so, what are they?
\item{C)}  Based on the snowfall depth data, describe how you could 
approximate the total accumulation of snow for the New England region
in the ``Blizzard of 1978.''  For the purposes of this, take 
this region to extend from $-74.0$ to $-70.0$ in longitude 
and $41.0$ to $45.0$ latitude.  (This includes a bit of New York,
and excludes most of Maine, but so be it!)  Note:  In this region,
a degree of latitude is about 69 miles and a degree of longitude
is about 51 miles.  There are $5280$ feet in a mile.  
\item{D)}  Carry out your approximation using the numerical data.
Note: Not all of the data corresponds to locations in the New 
England region that you want.  The small crosses in the map underlying
the density plot indicate the 
locations of the stations where depth measurements were made.
\item{E)}  What information would you need to make a {\it better}
approximation of the total snow accumulation? 
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