Math 166 Project
Area of Arizona

We want to estimate the area of Arizona using the information given on the map below.  We note that the area east of the dotted line can be approximated easily by computing the area of the rectangle whose left side is denoted by the dotted line and subtracting the area of the triangle at the southwest corner of the state.   The only difficult part of the estimate is in determining the area of the portion of the state west of the dotted line.  To estimate this area we want to use our knowledge
of calculus.

Map of Arizona

Note that we have placed the origin of our coordinate system at the southwest corner of the state.  With the coordinate system placed in this manner, the points indicated on the map have the following coordinates in miles:

Point A - (0,0)               Point F - (55.8,383)  
Point B - (55.8,0)            Point G - (55.8,346.25)
Point C - (223.25,0)          Point H - (55.8, 85.2)
Point D - (329,0)             Point I - (55.8,64.6)
Point E - (329,383)           Point J - (0,85.2 )

The points indicated by dots along the western portion of the state are given in the lists below.   The list datax represents the x-coordinate (using our coordinate system) of each point and the array datay represents the y-coordinate of each point.

Clear[datax,datay]

datax={0,1.7625,5.875,14.6875,20.5625,14.6875,5.875,11.75,
17.625,20.5625,35.8375,44.0625,29.375,26.4375,20.5625,17.625, 23.5,
20.5625,17.625,14.6875,17.5075,26.4375,38.1875,55.8125};  

datay={85.1875,94,102.813,102.93,111.625,123.375,126.313,146.64,
158.625,188,197.4,208.5632,217.375,237.937,249.688,261.437,
273.187,287.875,308.437,317.25,326.063,334.875,340.75,
346.625};

Clear[bdry,data,dots,cities,datax,datay,datax1,datay1,pts]
h=47;
xx=7.0625;
bot2=4.75*h;
bot3=bot2+(2.40)*h;
bot1=(1.1875)*h;
height=(8.32)*h;
h1=(1.375)*h;
datax1={xx-7.0625,xx-7.025,xx-6.9375,xx-6.75,xx-6.625,
xx-6.75,xx-6.9375,xx-6.8125,xx-6.6875,xx-6.625,xx-6.3,
xx-6.125,xx-6.4375,xx-6.5,xx-6.625,xx-6.6875,xx-6.5625,
xx-6.625,xx-6.6875,xx-6.75,xx-6.69,xx-6.5,xx-6.25,xx-5.875};
datax=h*datax1;
datay1={1.8125,2,2.1875,2.19,2.375,2.625,2.6875,3.12,3.375,4,
4.2,4.4375,4.625,5.0625,5.3125,5.5625,5.8125,6.125,6.5625,
6.75,6.9375,7.125,7.25,7.375};
datay=h*datay1; 
bdry=Line[ {{bot1,h*7.375}, {bot1,height}, {bot3,height},{bot3,0},
{bot2,0},{0,85.1875} }];
dots={{0,0},{bot2,0},{bot1,0},{bot1,height}};
p1=Show[Graphics[bdry],Graphics[{Dashing[{0.01,0.01}],
Line[dots]}] ]
data=Table[{datax[[i]],datay[[i]]},{i,1,24}];
cities={Point[{225,75}],Point[{120,140}],Point[{210,300}]};
p2=ListPlot[data,PlotJoined->True]
pts= Table[
 Point[{datax[[i]],datay[[i]]}],{i,1,24}]; 
Show[p1,p2,Graphics[{PointSize[.02],cities}],
Graphics[{PointSize[.015],pts}],
 Graphics[Text["A",{5,5},{-1,0}]], 
 Graphics[Text["B",{bot1+5,5},{-1,0}]],
 Graphics[Text["C",{bot2,5},{-1,0}]],
 Graphics[Text["D",{bot3-8,5},{-1,0}]],
 Graphics[Text["E",{bot3-8,height-5},{-1,0}]],
 Graphics[Text["F",{bot1+8,height-5},{-1,0}]],
 Graphics[Text["G",{bot1+5,h*7.375},{-1,0}]],
 Graphics[Text["H",{bot1+5,h*1.8125},{-1,0}]],
 Graphics[Text["I",{bot1+5,h1},{-1,0}]],
 Graphics[Text["J",{5,h*1.8125},{-1,0}]],
 Graphics[Text["Phoenix",{130,140},{-1,0}]],
 Graphics[Text["Tucson",{220,75},{1,0}]],
 Graphics[Text["Flagstaff",{200,300},{1,0}]], 
 Graphics[Text[FontForm["Arizona",{"Courier-Bold",18}],
 {140,200},{-1,0}]],PlotRange->All,AspectRatio->Automatic,
Axes->Automatic]

a.  Approximate the area of Arizona east of the dotted line.

rectangle=(329-55.8)*383

tri=0.5*(223.25-55.8)*64.6

areaeast=rectangle-tri

b.  Use the ListPlot command to plot the area west of the dotted line from point J to point G.

data=Table[{datay[[i]],55.8-datax[[i]]},{i,1,24}];
ListPlot[data,PlotJoined->True,AspectRatio->Automatic]

c.  To find the area west of the dotted line, first find the area of the triangle HIJ and then use a left Riemann sum and all the data to approximate the area west of the dotted line.  Using your result in (a), give an approximation to the total area of Arizona. ( Note that there may be some ambiguity as to which is the left or right Riemann sum.  For consistency, choose the height of the first rectangle in your left Riemann sum to be the height JH .)

hij=1/2*(85.2-64.6)*(55.8)

lrs=Sum[(55.8-datax[[i]])*(datay[[i+1]]-datay[[i]]),{i,1,23}]

areawestl=lrs+hij

areal=areaeast+areawestl

d.  Use the area of the triangle HIJ and a right Riemann sum using all the data to approximate the area west of the dotted line.  Using your result in (a), give an approximation to the total area of Arizona.

rrs=Sum[(55.8-datax[[i+1]])*(datay[[i+1]]-datay[[i]]),{i,1,23}]

areawestr=rrs+hij

arear=areaeast+areawestr

e.  The actual total area of Arizona is 114,002 square miles.  What is the  (i) the absolute error in your approximations from (c) and (d) amd (ii) the percentage of error in your approximations from (c) and (d)?

area=114002
Abs[arear-area]
Abs[areal-area]
100*Abs[arear-area]/area
100*Abs[areal-area]/area