# Estimating the Area of Virginia

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

Integration is an important calculus topic that has a variety of applications.  Area is one of the most important applications of the definite integral.  However, the boundaries of some regions are irregular, and it is therefore difficult to create a definite integral that will closely approximate the region's area.  The state of Virginia is one such region.

In this project you will use Riemann sums and the computer algebra system MAPLE to begin your investigation of  the area of a complex region.  More specifically you will mark boundary points on a map of Virginia (excluding the Eastern Shore) and then import their coordinates to a MAPLE worksheet.  You will then learn the MAPLE commands for plotting these points and the rectangles, trapezoids, and parabolas they define.  In addition you will use MAPLE to find the area of these sub-regions and to approximate the total area.

For the Instructor

 Julie Clark, Caren Diefenderfer, and Trish Hammer are in the Department of Mathematics and Statistics at Hollins University.  Steve Hammer is in the Department of Mathematics at Virginia Western Community College.

# Estimating the Area of Virginia - Introduction

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

Integration is an important calculus topic that has a variety of applications.  Area is one of the most important applications of the definite integral.  However, the boundaries of some regions are irregular, and it is therefore difficult to create a definite integral that will closely approximate the region's area.  The state of Virginia is one such region.

In this project you will use Riemann sums and the computer algebra system MAPLE to begin your investigation of  the area of a complex region.  More specifically you will mark boundary points on a map of Virginia (excluding the Eastern Shore) and then import their coordinates to a MAPLE worksheet.  You will then learn the MAPLE commands for plotting these points and the rectangles, trapezoids, and parabolas they define.  In addition you will use MAPLE to find the area of these sub-regions and to approximate the total area.

For the Instructor

 Julie Clark, Caren Diefenderfer, and Trish Hammer are in the Department of Mathematics and Statistics at Hollins University.  Steve Hammer is in the Department of Mathematics at Virginia Western Community College.

# Estimating the Area of Virginia - Learning to Use the Virginia State Map

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

To complete this project you will need to understand how to use the interactive map of Virginia shown below.

Using this Live Map
• Click any place on the map to move the cursor to that point.

• Notice the readout at the upper right gives the location of the cursor measured in pixels from the lower left corner of the map.

• You can fine tune the location of the cursor by clicking on the four arrows at the left, right, top, and bottom of the map. Each click moves the cursor one pixel in the direction indicated by the arrow.

Questions:

1. Find the coordinates of Roanoke and of Richmond.

2. For our purposes, we will assume the coordinates of the southwest corner of Virginia are (24,98).  Locate this point on the map.

3. Click the blue Mark point button to create a point on the map at these coordinates.

4. Mark any 5 additional points on the northern state boundary.  Record the coordinates of these points.

5. Assume the coordinates of the southeast corner are (474, 98).  Mark this point on the map.

6. Click the red List points button.  (You may need to enlarge the window that opens up.)  Compare the values that you recorded to the ones in this list.  Can you see how they are the same?

7. Click the black Clear points button, and then repeat steps 2-6 until you are comfortable with this process.

Sources:

The map used here was obtained at http://g-lea.tamu.edu/smap.htm.

The applet used here is described in the Lite Applets article and can be downloaded from the resources page at the end of the article.

# Estimating the Area of Virginia - Using Left Rectangles

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

1.  Getting Started

Click the button at the right to open a MAPLE worksheet entitled leftstudent.mws.  If you are given a choice, you should save the file to your preferred directory, then navigate to that directory and open the file from there.  In the MAPLE worksheet, position your cursor anywhere in the line  [ > restart ;   and press Enter.   Pressing the Enter key executes the MAPLE code on the current line.  Activating the restart MAPLE command will clear all MAPLE variables, and it is important to do this whenever you start a new MAPLE project.

Now resize your MAPLE and browser windows so that you can see them both, side-by-side. Click in either window to make it the active window.

Your screen should look something like this:

2.  Collection of Data

Notice that you need to put the coordinates of points from a live map of Virginia into the MAPLE worksheet.  We will use the Virginia map you saw earlier to find and mark these points.  Click the picture at the right to learn how to get the point coordinates and store your coordinates in the MAPLE worksheet.

3.  Visualization of Data

Now that you have entered your x and y coordinates, you can work through the MAPLE worksheet by pressing the Enter key to execute the MAPLE commands.  (Note:  The MAPLE window must be the active window.)  The first output you see should be a red horizontal line (the southern border of Virginia) and eleven black points (the eleven, evenly spaced boundary points that you determined in section 2).  You should be able to "see" that this line and eleven points give a rough outline of the state of Virginia.  Repeat sections 2 and 3 until you are satisfied that you have obtained a reasonable rough outline of the state of Virginia.

4.  Construction of Approximating Left Rectangles

This section includes the MAPLE commands that construct our left rectangle approximation on each of the ten subintervals. Press the Enter key to execute the block of MAPLE commands.

5.  Visualization of Approximating Left Rectangles

This section begins with a loop of MAPLE code that defines the vertical partition lines and ends with a display of the ten approximating rectangles.  Does the red outline look like Virginia?  Compare your picture with the one at the top of this page. Note that the rectangular regions extend from the northern red boundary of Virginia to the x-axis.

6.  Area Calculation by Geometric Formula

MAPLE commands in this section use the geometric formula to calculate the area of the rectangles. (Note that each of these rectangles extends to the x-axis.)  This sum includes the area of the red region (the state of Virginia) as well as the area of the rectangular region between the southern boundary of Virginia and the x-axis.  Thus, our left endpoint approximation (in square pixels) must be adjusted.

7.  Solution

We must now convert square pixels to square miles.  Use the map scale to determine how many pixels are equivalent to 80 miles.  In the MAPLE worksheet, determine the conversion factor from square pixels to square miles and use this factor to determine the rectangular approximation (using 10 subintervals) for the area of Virginia.

8.  Left Endpoint Summary

The image at the top of this page shows the left endpoint approximation superimposed on the map of Virginia.  Use this image to determine whether this approximation using 10 subintervals is an over- or under- approximation.  Explain your reasoning and include your response in the text cell provided in the MAPLE worksheet.  Save your worksheet as left10****.mws. (Replace the **** with your first initial and last name.)

### Assignment

The purpose of this assignment is to increase the accuracy of our approximation by using more rectangles.  In this assignment, you will repeat the above process based on the collection of more data points.

Step 1:  Open  your left10****.mws worksheet and save it as left30****.mws.  (You now have two copies of the same worksheet.)

Step 2:  Use a live map of Virginia to collect 31 evenly spaced points on the northern boundary of Virginia.  Your first point must be (24, 98) and the last point must be (474, 98).  Paste these coordinates into your left30****.mws worksheet.

Step 3:  Repeat sections 3-8 (above) for your new data points by modifying your left30 worksheet.

# Estimating the Area of Virginia - Using Right Rectangles

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

1.  Getting Started

Click the button at the right to open a MAPLE worksheet entitled rightstudent.mws.  If you are given a choice, you should save the file to your preferred directory, then navigate to that directory and open the file from there.  In the MAPLE worksheet, position your cursor anywhere in the line  [ > restart ;   and press Enter.   Pressing the Enter key executes the MAPLE code on the current line.  Activating the restart MAPLE command will clear all MAPLE variables, and it is important to do this whenever you start a new MAPLE project.

Now resize your MAPLE and Internet Explorer windows so that you can see them both, side-by-side. Click in either window to make it the active window.

Your screen should look something like this:

2.  Collection of Data

Notice that the MAPLE worksheet needs your input of the coordinates of points from a map of Virginia.  We will use the Virginia map you saw earlier to find and mark these points.  Click the picture at the right to learn how to get the point coordinates and store your coordinates in the MAPLE worksheet.

3.  Visualization of Data

Now that you have entered your x and y coordinates, you can work through the MAPLE worksheet by pressing the Enter key on your computer to execute the MAPLE commands.  (Note:  The MAPLE window must be the active window.)  The first output you see should be a red horizontal line (the southern border of Virginia) and eleven black points (the  eleven, evenly spaced boundary points that you determined in section 2).  You should be able to "see" that this line and eleven points give a rough outline of the state of Virginia.  Repeat sections 2 and 3 until you are satisfied that you have obtained a reasonable rough outline of the state of Virginia.

4.  Construction of Approximating Right Rectangles

This section includes the MAPLE commands that construct our right rectangle approximation on each of the ten subintervals.  Press the Enter key to execute the block of MAPLE commands.

5.  Visualization of Approximating Right Rectangles

This section begins with a loop of MAPLE code that defines the vertical partition lines and ends with a display of the ten approximating rectangles.  Does the red outline look like Virginia?  Compare your picture with the one at the top of this page. Note that the rectangular regions extend from the northern red boundary of Virginia to the x-axis.

6.  Area Calculation by Geometric Formula

MAPLE commands in this section use the geometric formula to calculate the area of the rectangles. (Note that each of these rectangles extends to the x-axis.)  This sum includes the area of the red region (the state of Virginia) as well as the area of the rectangular region between the southern boundary of Virginia and the x-axis.  Thus, our right endpoint approximation (in square pixels) must be adjusted.

7.  Solution

We must now convert square pixels to square miles.  Use the map scale to determine how many pixels are equivalent to 80 miles.  In the MAPLE worksheet, determine the conversion factor from square pixels to square miles and use this factor to determine the rectangular approximation (using 10 subintervals) for the area of Virginia.

8.  Right Endpoint Summary

The image at the top of this page shows the right endpoint approximation superimposed on the map of Virginia.  Use this image to determine whether this approximation using 10 subintervals is an over- or under- approximation.  Explain your reasoning and include your response in the text cell provided in the MAPLE worksheet.  Save your worksheet as right10****.mws. (Replace the **** with your first initial and last name.)

### Assignment

The purpose of this assignment is to increase the accuracy of our approximation by using more rectangles.  In this assignment, you will repeat the above process based on the collection of more data points.

Step 1:  Open  your right10****.mws worksheet and save it as right30****.mws.  (You now have two copies of the same worksheet.)

Step 2:  Use a live map of Virginia to collect 31 evenly spaced points on the northern boundary of Virginia.  Your first point must be (24, 98) and the last point must be (474, 98).  Paste these coordinates into your right30****.mws worksheet.

Step 3:  Repeat sections 3-8 (above) for your new data points by modifying your right30 worksheet.

# Estimating the Area of Virginia - Using Trapezoids

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

1.  Getting Started

Click the button at the right to open a MAPLE worksheet entitled trapstudent.mws.  If you are given a choice, you should save the file to your preferred directory, then navigate to that directory and open the file from there.  In the MAPLE worksheet, position your cursor anywhere in the line  [ > restart ;   and press Enter.   Pressing the Enter key executes the MAPLE code on the current line.  Activating the restart MAPLE command will clear all MAPLE variables, and it is important to do this whenever you start a new MAPLE project.

Now resize your MAPLE and browser windows so that you can see them both, side-by-side. Click in either window to make it the active window.

Your screen should look something like this:

2.  Collection of Data

Notice that the MAPLE worksheet needs your input of the coordinates of points from a map of Virginia.  We will use the Virginia map you saw earlier to find and mark these points.  Click the picture at the right to learn how to get the point coordinates and store your coordinates in the MAPLE worksheet.

3.  Visualization of Data

Now that you have entered your x and y coordinates, you can work through the MAPLE worksheet by pressing the Enter key on your computer to execute the MAPLE commands.  (Note:  The MAPLE window must be the active window.)  The first output you see should be a red horizontal line (the southern border of Virginia) and eleven black points (the  eleven, evenly spaced boundary points that you determined in section 2).  You should be able to "see" that this line and eleven points give a rough outline of the state of Virginia.  Repeat sections 2 and 3 until you are satisfied that you have obtained a reasonable rough outline of the state of Virginia.

4.  Construction of Approximating Trapezoids

This section includes the MAPLE commands that define the 10 linear functions we will use to construct our trapezoidal approximation. Press the Enter key to execute the block of MAPLE commands.

5.  Visualization of Approximating Trapezoids

This section begins with a loop of MAPLE code that defines the vertical partition lines and ends with a display of the ten approximating trapezoids.  Does the red outline look like Virginia?  Compare your picture with the one at the top of this page. Note that the trapezoids extend from the northern red boundary of Virginia to the x-axis.

6.  Area Calculation Using Geometric Formulas

These MAPLE commands compute the sum of the area of the individual trapezoids.  (Each of these trapezoids extends to the x-axis.)  This sum includes the area of the red region (the state of Virginia) as well as the area of the rectangular region between the southern boundary of Virginia and the x-axis.  Thus, our trapezoidal approximation (in square pixels) must be adjusted.

7.  Area Calculation Using Integrals

MAPLE commands in this section use integrals to compute the sum of the area of the individual trapezoids.  (Note:  Integration is not necessary when using trapezoids, but it will be necessary inthe next section when we use parabolas in Simpson's rule.)

8.  Solution

We must now convert square pixels to square miles.  Use the map scale to determine how many pixels are equivalent to 80 miles.  In the MAPLE worksheet, determine the conversion factor from square pixels to square miles and use this factor to determine the trapezoidal approximation (using 10 subintervals) for the area of Virginia.

9.  Trapezoidal Summary

The image at the top of this page shows the trapezoidal approximation superimposed on the map of Virginia.  Use this image to determine whether this trapezoidal approximation using 10 subintervals is an over- or under- approximation.  Explain your reasoning and include your response in the text cell provided in the MAPLE worksheet.  Save your worksheet as trap10****.mws. (Replace the **** with your first initial and last name.)

### Assignment

The purpose of this assignment is to increase the accuracy of our approximation by using more trapezoids.  In this assignment, you will repeat the above process based on the collection of more data points.

Step 1:  Open  your trap10****.mws worksheet and save it as trap30****.mws.  (You now have two copies of the same worksheet.)

Step 2:  Use a live map of Virginia to collect 31 evenly spaced points on the northern boundary of Virginia.  Your first point must be (24, 98) and the last point must be (474, 98).  Paste these coordinates into your trap30****.mws worksheet.

Step 3:  Repeat sections 3-9 (above) for your new data points by modifying your trap30 worksheet.

# Estimating the Area of Virginia - Using Simpson's Rule

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

1.  Getting Started

Click the button at the right to open a MAPLE worksheet entitled simpstudent.mws.  If you are given a choice, you should save the file to your preferred directory, then navigate to that directory and open the file from there.  In the MAPLE worksheet, position your cursor anywhere in the line  [ > restart ;   and press Enter.   Pressing the Enter key executes the MAPLE code on the current line.  Activating the restart command will clear all MAPLE variables, and it is important to do this whenever you start a new MAPLE project.

Now resize your MAPLE and browser windows so that you can see them both, side-by-side. Click in either window to make it the active window.

Your screen should look something like this:

2.  Collection of Data

Notice that the MAPLE worksheet needs your input of the coordinates of points from a map of Virginia.  We will use the Virginia map you saw earlier to find and mark these points.  Click the picture at the right to learn how to get the point coordinates and store your coordinates in the MAPLE worksheet.

3.  Visualization of Data

Now that you have entered your x and y coordinates, you can work through the MAPLE worksheet by pressing the Enter key on your computer to execute the MAPLE commands.  (Note: The MAPLE window must be the active window.)  The first output you see should be a red horizontal line (the southern border of Virginia) and eleven black points (the  eleven, evenly spaced boundary points that you determined in section 2).  You should be able to "see" that this line and eleven points give a rough outline of the state of Virginia.  Repeat sections 2 and 3 until you are satisfied that you have obtained a reasonable rough outline of the state of Virginia.

4.  Construction of Approximating Parabolas

This section includes the MAPLE commands that define the 5 quadratic functions we will use to construct our Simpson's rule approximation. Press the Enter key to execute the block of MAPLE commands.

5.  Visualization of Approximating Parabolas

This section begins with a loop of MAPLE code that defines the vertical partition lines and ends with a display of the five approximating parabolas.  Does the red outline look like Virginia?  Compare your picture with the one at the top of this page . Note that the parabolic regions extend from the northern red boundary of Virginia to the x-axis.

6.  Area Calculation by Integration

These MAPLE commands compute the sum of the areas between the approximating parabolas and the x-axis.   This sum includes the area of the red region (the state of Virginia) as well as the area of the rectangular region between the southern boundary of Virginia and the x-axis.  Thus, our Simpson's rule approximation (in square pixels) must be adjusted.

7.  Area Calculation by Simpson's Rule Formula

MAPLE commands in this section use the Simpson's rule formula (from your textbook) to calculate the area between the parabolic curves and the x axis.

8.  Solution

We must now convert square pixels to square miles.  Use the map scale to determine how many pixels are equivalent to 80 miles.  In the MAPLE worksheet, determine the conversion factor from square pixels to square miles, and use this factor to determine the Simpson's rule approximation (using 10 subintervals) for the area of Virginia.

9.  Simpson's Rule Summary

The image at the top of this page shows the Simpson's rule approximation superimposed on the map of Virginia.  Use this image to determine whether this approximation using 10 subintervals is an over- or under- approximation.  Explain your reasoning, and include your response in the text cell provided in the MAPLE worksheet.  Save your worksheet as simp10****.mws. (Replace the **** with your first initial and last name.)

### Assignment

The purpose of this assignment is to increase the accuracy of your approximation by using more parabolas.  In this assignment, you will repeat the above process based on the collection of more data points.

Step 1:  Open  your simp10****.mws worksheet and save it as simp30****.mws.  (You now have two copies of the same worksheet.)

Step 2:  Use a live map of Virginia to collect 31 evenly spaced points on the northern boundary of Virginia.  Your first point must be (24, 98) and the last point must be (474, 98).  Paste these coordinates into your simp30****.mws worksheet.

Step 3:  Repeat sections 3-9 (above) for your new data points by modifying your simp30 worksheet.

# Estimating the Area of Virginia - Project Assignment

Author(s):
Julie Clark, Caren Diefenderfer, Steve Hammer, and Trish Hammer,

 California Illinois Maryland North Carolina

Choose any one of these states and use MAPLE to approximate its area in two ways:

1.  Use a trapezoidal approximation with at least 20 subintervals

2.  Use Simpson's rule (parabolas) with at least 20 subintervals.