MATH 235 Lab #4:

Riemann Sums and the Area Under a Curve

Directions:

In this lab, we will generate important conclusions about Riemann sums and areas under a curve. The conclusions must be memorized and will appear on exams. "Area under a curve" means the region bounded by the -axis and the given function. The function can be above the -axis, below the -axis, or a combination of both. If the function is below the -axis, the area between the axis and the function will be represented by a negative number.

Format: Solutions should be summited as a lab report. A "lab report" means that more explanation should be provided compared to a homework assignment. Your name should be located at the top of your report as well as Lab #4. Then, you should write the problem number and solution. Your solution should include all the work required to answer the question including explanations in complete sentences. Problems should be presented in numerical order. You may choose to typeset or hand-write your solutions, but note that there are places where you will have to include graphics from Geogebra in your work. Do not provide several pages of hand written work with all Geogebra graphics at the end. The report needs to be professional and easy to follow.

Grading: This lab is worth 100 points. Details are provided below.

Formatting: 3 points2a) 14 points

4-graph

Using 5 decimal places: 5 points2-sentence to explain graph

5-areas with notation

1a & b) 22 points1-talk about under/over-estimate

4-graph2-explanation

2-sentence to explain graph

2-area with notation2b) 14 points

1-state under/over-estimate4-areas and notation

1-explain why4-graph

1-state incr/decr2-sentence to explain graph

2-conclusion statement

1c)8 points2-explanation

2-conclusion statement

6-completed table3a) 7 points

2-sentence about function

1d) 11 points3-answer with notation & units

7-areas in a sentence with notation2-explanation

2-conclusion statement

2-explanation3b & c) 16 points

4-graph

2-sentence to explain graph

2-answer with notation

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Calculus Overview

We have seen that the area under a velocity function on a given interval gives the change in position (if the function is above and below the -axis) or the total distance traveled (if the function is always above the -axis).

Conclusion 1: So far the EXACT area under the curve has been computed by dividing the region into rectangles, triangles, and circles, which are familiar geometric shapes. (This won't work for every velocity function as shown in the graph of below and on the left.) One way to APPROXIMATE the area is to divide the region into rectangles whose height is determined by the graph. This is called a Riemann sum.

This lab explores Riemann sums numerically and graphically. The middle graph above shows 4 rectangles used to approximate the area whose right sides intersect the graph. In other words, the right side determines the height of the rectangle for each subinterval. We use the notation R(4) to denote this Riemann sum and call it a right Riemann sum. Geogebra also gives the numerical value of the sum on the graph as "a=4.69" so we write R(4) = 4.69.

The graph on the right shows 4 rectangles whose left sides intersect the graph, meaning that the left side determines the height of each rectangle. We use the notation L(4) = 5.08 and call this a left Riemann sum.

It is possible to use as many rectangles as you like to approximate the area under the curve. Note that Geogebra calculates areas under the -axis as negatives.

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Geogebra Overview

To create graphs like the ones above, we do the following in Geogebra:

1.)Type the function into the input line and press enter.

2.)Type the Geogebra command RectangleSum into another input line, using the syntax provided below, and press enter.

RectangleSum( , , , , )

The "Position for rectangle start" tells the program whether you want the left or right side of each rectangle to intersect the graph as follows:

for a left rectangle sum, use "0"

for a right rectangle sum, use "1"

The numerical value of the sum of the areas of the rectangles automatically appears on the graph. Be sure that you set the rounding option to 5 decimal places for this report.

Example: To graph the function and approximate the area under the curve on the interval using 5 rectangles with the right side of the rectangle determining the heights, we use the following two commands in separate input lines:

rectanglesum( f(x),0.4,3.2,5,1)

Both commands must be used every time you are asked to determine an area.

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Setup for the Problems

The functions used in each problem contain a constant and/or so that each student works with a slightly different function. These constants are based on your birthday. The represents the number of the month you were born, while stands for the day of the month you were born.

For example, if your birthday is March 19^th, then and . It is important for the professor to know what numbers you are using for and . Do not use the letters and in your report. Type the numbers they represent instead.

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Problems

1. Answer the following questions about the function on the interval . (Remember that you need to replace with the correct number from your birthday.)

a.Approximate the area under on the given interval by using 10 rectangles whose left sides determine the heights. Include a copy of your work from Geogebra. Adjust the numbers on the axes so that you provide the best view of the graph and rectangles as possible.

To explain this graph, write the following without the quotation marks:

"The graph shows the function (type the number for ) and its Riemann sum using 10 rectangles with left endpoints on the interval ."

Then state the area, i.e. Riemann sum, using the appropriate notation discussed in the Calculus Overview. Does the Riemann sum appear to be an under- or over-estimate for the actual area under the curve? Why? What behavior of the function makes this happen? (Hint: Is the function increasing or decreasing?)

b.Approximate the area under on the given interval by using 10 rectangles whose right sides determine the heights. (Note that you don't need to retype any commands. Just change the last number in the rectanglesum command to be 1.) Include a copy of your work from Geogebra. Adjust the numbers on the axes so that you provide the best view of the graph and rectangles as possible.

Explain the graph using sentences similar to the ones in part a. State the area, using appropriate notation and explain whether this is an under- or over-estimate for the actual area under the curve. What behavior of the function makes this happen? (Hint: Is the function increasing or decreasing?)

c.It is possible to determine whether a Riemann sum is an over- or under-estimate of the actual area under the curve based on whether the function is increasing or decreasing on the given interval. The goal of this problem is to generate a conclusion that summarizes these results. After completing the following, state over-estimate or under-estimate in each of the 4 boxes within the table.

Explain this information by writing the following without the quotation marks:

"Conclusion 2: A Riemann sum represents an over- or under-estimate of the actual area under a curve based on whether the function is increasing or decreasing on the given interval. The table below summarizes the results. L(n) represents the Riemann sum using n rectangles whose heights are determined by the left side of the subinterval." Now write a similar sentence describing what R(n) represents. Use the results from parts a and b to complete two of the boxes. Use Geogebra to draw . Don't retype the function. Just type in an input line. Now use Geogebra to draw R(10) for . In your rectanglesum command just type a negative in front of and press enter. Note whether is increasing and whether R(10) is an over- or under-estimate. Write your result in the table. You do not need to include a graph or the area. Finally, draw L(10) for . In your rectanglesum command just change the last number to be 0. Note whether is increasing and whether L(10) is an over- or under-estimate. Write your result in the table.

Increasing function

Decreasing function

L(n)

R(n)

d.Now we explore what happens when we use more rectangles. Compute R(50), R(100), L(50), and L(100) for . Write the answers in a sentence using appropriate notation. Do not include the pictures in your report but study them carefully. Then write "Conclusion 3: based on the graphs of Riemann sum rectangles, a more accurate estimate of the area under a curve is obtained from using (use the word more or less) rectangles." Explain your answer.

2. Answer the following questions about the function on the interval . (Remember that you need to replace with the correct number from your birthday. In this problem, if you were born in January, use instead of .)

a.Approximate the area under on the given interval by using 25 rectangles whose left sides determine the heights AND by 25 rectangles whose right sides determine the heights. Include a copy of one of the graphs from Geogebra. Adjust the numbers on the axes so that you provide the best view of and the rectangles as possible. Explain the graph by using similar wording provided in quotation marks in problem #1a. Note that it will be necessary to change the function, the number of rectangles, and the interval. Then state the areas using the appropriate notation in a sentence. Can you tell whether the Riemann sum is an under- or over-estimate of the actual area under the curve in either case? Explain. Hint: Think about Conclusion 1.

b.It is also possible to calculate a Riemann sum where the height of the rectangle is determined using the midpoint of each rectangle. This is done using the rectanglesum command with 0.5 for the "Position for rectangle start" instead of 0 or 1. The notation for this, when we use 25 rectangles, is M(25). Compute M(25) and M(75) and state these areas. Include a copy of the graph of M(75) from Geogebra. Adjust the numbers on the axes so that you provide the best view of the graph and rectangles as possible. Explain the graph by using similar wording provided in quotation marks in problem #1a. Then write: "Conclusion 4: Riemann sums using (include the word left, right, or midpoints) to determine the height of the rectangles seems to provide the most accurate estimate of the area." Explain why this is true.

3. Suppose that the velocity of a moving object is given by the function where is measured in seconds and is measured in feet per second. (Hint: This function is in terms of . In Geogebra, you can use either or , but you can't use both in the same equation. Be consistent. Be sure to type a multiplication * between and . Otherwise a slider might appear.) The goal is to estimate the distance traveled by the object on the interval as accurately as possible.

a.Draw a variety of Riemann sums using Geogebra. Use left, right, and midpoints for the height of the rectangles. Hint: You might want to start with 1000 rectangles. Include a sentence stating your function, what it represents, and the time interval provided. Report your best estimate of the distance traveled, using appropriate notation to illustrate what method and the number of rectangles used, and include units of measurement on the answer. Explain why you think this is the best estimate. You do not need to provide a graph of your work.

b.Find the LEAST number of rectangles that will achieve your answer in part a using midpoints to determine the height of the rectangles. Include a copy of your Geogebra graph. Adjust the numbers on the axes so that you provide the best view of the graph and rectangles as possible. Explain the graph by using similar wording provided in quotation marks in problem #1a. State the area using appropriate notation.

c.Find the LEAST number of rectangles that will achieve your answer in part a using rectangles whose left sides determine the heights. Include a copy of your Geogebra graph. Adjust the numbers on the axes so that you provide the best view of the graph and rectangles as possible. Explain the graph by using similar wording provided in quotation marks in problem #1a. State the area using appropriate notation.