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4 RIEMANN SUMS

ÇİĞDEM ÖZDEMİR

OBJECTIVES

CCSS.MATH.CONTENT.HSA MODELING

MoNE: İD.12.2.1.1

MATERIALS

Graphing display calculator

Computer with word processing software (i.e. Microsoft Word)

GeoGebra software

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ACTIVITY 1

Introduction

Suppose that you have just inherited a land in Ankara between Eskişehir Yolu and Sakıp Sabancı Bulvarı, and that this land is located between the CD and EF lines.

Questions

1) How would you determine the area of your land? Write your thoughts.

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2) See the picture below. If you estimate the area of your land to be the area of the DFEG rectangle, this may not be the best estimate. You may be missing the circled area in left top corner or you may be incorrectly including the shaded area. How could you estimate this better?

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3) See the picture below. If you calculate the area of your land to be the sum of two rectangles with the same width above, you would have a better estimation, but you would still miss some areas and include other areas. What if you were to divide your property into three parts? Which one would provide you with a better estimation?

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ACTIVITY 2

Instructions

An approximation of an area under a curve which is made by using rectangles is known as Riemann Sums.

In order to find the real approximate area of your land, go to Google Maps, and find the distance between two points by clicking on “Get Directions.”

As shown in the picture below, the distance between A and B is 650 meters. Therefore, we can say that the distance between C and D is approximately 650 meters as well.

Now let’s make these estimations mathematically. The Sakıp

Sabancı Bulvarı is , Eskişehir yolu is the x-axis and CD line is the y-axis. You can use GeoGebra software to find the corresponding equation for the function of Sakıp Sabancı Bulvarı.

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As you see from the picture above, the approximate function for

Sakıp Sabancı Bulvarı is:

20.04 0.82 6.81f x x x

Please pay attention to the scale! In this picture, 650 m corresponds with approximately 6.5 cm on Geogebra

Now that you know the function and your x interval, which is (0,20), you can use your calculator to easily view the Riemann sums either with left rectangles, right rectangles, or trapezoids. First, you should download the RIEMANN program from the link below and send the file to your calculator.

http://www.calcblog.com/?download=riemann

To begin, enter the function in the Y1 variable by pressing [Y=]

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Press [PRGM], select RIEMANN, and press [ENTER] to run.

If you have your function in Y1, enter #1 to continue, then enter #1 to draw a graph or #2 to solve numerically.

For this problem, enter your left endpoint as 0 and press [ENTER], then enter your right endpoint as 20 and press [ENTER].

To calculate the left rectangle, press #1 and press [ENTER], or enter the number that corresponds with your choice. Then enter the number of slices. For this problem, select the right rectangle and then enter 2 slices for the first step.

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Your graph will appear as shown in the picture below.

Press [ENTER] to see the sum of your rectangles.

Press [ENTER] to change your choice or slice number.

Now you can change the slice number by pressing [ENTER] when 1: is highlighted

Enter 10 for N and press [ENTER]

Your graph will appear as shown in the picture below.

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Press [ENTER] to see the sum of the rectangles.

Now, change your slice number to 200. Verify that your graph appears like the picture shown below and that the sum of the rectangles is 78.888.

Attention!!! Taking the scale into account, you will have to multiply your result by 10000.

So with 200 slices, the area of your land is approximately

278.888 10000 788880m

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Questions

4) Compare your findings and discuss which one offers the best estimation.

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5) Find a better estimation of the area of your land by changing the slice number.

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ACTIVITY 3

Instructions

Press [MATH] and choose fnInt (using the ▼ arrow, or just press #9 and press [ENTER].

For the left endpoint, enter 0 and for the right endpoint, enter 20 by moving the arrows.

For the inside parenthesis, press [VARS] and move Y-VARS using the right arrow.

Press [ENTER] if 1: is highlighted.

Choose Y1 if your function is located in the Y1 variable.

Then press [X,T,ø,n], select “x” for dx, and press [ENTER] to

calculate the integral.

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Questions

6) Find the definite integral of your function between 0 and 20 using your graphing calculator:

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7) To what degree are your Riemann sums findings approximate to the definite integral?

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ACTIVITY 4

Instructions

We can find the Riemann sums numerically with Excel as well. Open a document in Excel. For our example, since the length of our interval is 20 and we want 20 subintervals, the width of each subinterval is x=1 cm. In fact, it's not hard to see that we will want to have the x values:

x = 1, x = 2, x = 3, . . . , x = 20.

To input these values into the worksheet, type 1 into cell A1 and 2 into cell A2. Then use the mouse to highlight cells A1 and A2.

Next click on the small black square in the lower right corner of the highlighted box and drag column A down to cell A21. This will fill column A with numbers 1 to 21.

To evaluate our function at all these points along the x-axis, enter the function in cell B1. To do this, highlight cell B1, type

20.04 1 0.82 1 6.81A A and press [ENTER].

Cell B1 should now show the value of our function at x=1. (The value is 6.03).

Next, highlight cell B1 and then click on the small black square in the lower right corner, and drag the column down to cell B21. This will fill column B with the values of our function at the points named in column A.

The left Riemann sum is equal to 1 times the sum of the entries in cells B1 through B20. To compute this, select an empty cell, highlight it, and then enter:

=1*TOPLA(B1:B20) for Turkish or,

=1*sum(B1:B20) for English

and press [ENTER].

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We can enter an identifier for this number, such as LeftSum:, in an empty cell.

The right Riemann sum is equal to 1 times the sum of the entries in cells B2 through B21. To compute this, select another empty cell, highlight it, and then type

=1*sum(B2:B21)

and press [ENTER].

Below is how the worksheet will appear when we have finished:

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Questions

10) Change the numbers for your subintervals and find a better estimate area of your land using Excel.

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11) Verify that you achieve the same results as with your graphing calculator.

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