Left Endpoint Approximations

1.  Getting Started

First, we clear all variables in Maple.

>    restart;

2. Collection of Data

Next, we enter the data points from the border of Virginia.  We have identified 11 data points (and consequently a regular partition of 10 subintervals).

>    xborder:=[24,69,114,159,204,249,294,339,384,429,474]:

>    yborder:=[98,122,160,152,160,217,243,298,282,211,98]:

3.  Visualization of Data

We create a plot of the data points

>    borderdata:=PLOT(POINTS([xborder[t],yborder[t]] $t=1..11)):

and also a plot of a horizontal line for the southern border.

>    sborder:=plot(98,x=xborder[1]..xborder[11],color=red,thickness=2):

And now display the data points and the southern border.

>    with(plots):

Warning, the name changecoords has been redefined

>    display(borderdata,sborder,view=[0..480,0..300]);

[Maple Plot]

Do the data points and southern border provide a rough outline of the state of Virginia?


4.  Construction of Approximating Left Rectangles

We construct an approximating left endpoint rectangle on each of the 10 subintervals. The following loop creates a plot of the upper edge of each of these rectangles.

>    for i from 1 to 10 do

>    pplotred[i]:=plot(yborder[i],x=xborder[i]..xborder[i+1],color=red,thickness=2):

>    od:


5.  Visualization of Approximating Left Rectangles

To see the left rectangles,  we draw in vertical lines corresponding to our 10 subintervals. The following commands create a plot of these partition lines (note that there are 11 partition lines).

>    partition[1]:=PLOT(CURVES([[xborder[1],0],[xborder[1],yborder[1]]])):

>    for i from 2 to 11 do

>    partition[i]:=PLOT(CURVES([[xborder[i],0],[xborder[i],max(yborder[i-1],yborder[i])]])):

>    OD:

>    partitionlines:=display(partition[k] $k=1..11):

Now, we display what we've created so far.

>    display(sborder,borderdata,partitionlines,pplotred[t] $t=1..10,view=[0..480,0..310]);

[Maple Plot]


Does the region bounded above by the rectangles and below by the red horizontal line look like Virginia?

Note that the left rectangles extend from the northern border of Virginia to the x axis.

6.  Area Calculation by Geometric Formula

To find the area between the northern rectangular border and the x axis, we use a geometric formula to determine the area of each of the left rectangles on each of the 10 subintervals and then sum the areas. Note that this sum includes the area of Virginia as well as the area of the rectangular region between the southern border of Virginia and the x axis.  So, this approximation must be adjusted.

>    sum(45*yborder[j],j=1..10);


To calculate the area of the red outlined Virginia, we must subtract the area of the rectangle with top edge defined by the southern border of Virginia

>    87435-(474-24)*98;


Thus, the left endpoint approximation (using 10 subintervals) of the area of Virginia (minus the Eastern Shore) is 43335 square pixels.


7.  Solution

According to the map scale,  80 miles = 87 pixels  so 80/87  miles = 1 pixel    so (80/87)^2  square miles = 1 square pixel

We convert 43335 square pixels to square miles

>    43335.*(80/87)^2;




We conclude that the left endpoint approximation(using 10 subintervals) of the area of Virginia (minus the Eastern Shore) is 36642 square miles.


8.  Left Endpoint Summary

Is your left endpoint  approximation an under or over approximation?  Explain.