Math 231 assignment

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Math 2231 Name: _____________________________

Project 2

Area Approximations

This project involves investigations of Riemann Sums and Integrals. The project should be

organized and submitted in the following order (1 pt):

1. This packet without the excel instructions (only the first 7 pages of this packet, please don’t submit the excel directions with your project). Take pictures or scan these pages

and submit them in blackboard

2. Desmos.com colored graphs in the order of n = 3, n=12 (submit the link to your graph in blackboard).

3. Your excel file should be submitted in blackboard by the due date as well.

Late projects will lose 10 points per day, including weekends.

As you complete the Excel spreadsheets, fill in the table below (some boxes will be blank, because some approximations are not completed for all values of n): Round to 6 decimal places when possible in the table (worth 2 points total for the table).

Approximations Left Endpoint Right Endpoint

Midpoint Trapezoidal Rule

Simpson’s Rule

n = 4

n = 6

n = 12

n = 24

n = 50

n = 100

n = 200

n = 500

n = 1000

Part 1: Exploration of Rectangle Approximations: Left Endpoints

We will be exploring the graph of f(x) = 𝑥3

3 −

7𝑥2

3 +

10𝑥

3 + 4. Using LEFT ENPOINTS,

approximate the area of the region between the graph of f(x) and the x-axis on the interval [0, 6] with n = 3, n = 6, n = 12. Use subintervals of equal width. Show your work, using sigma notation where necessary. Round to 2 decimal places if necessary.

1. (4 pts) Numerical Interpretation

• n = 3

• n = 6

• n = 12

2. (3 pts) Graphical Representation

Using desmos.com, graph the function f(x) = 𝑥3

3 −

7𝑥2

3 +

10𝑥

3 + 4. Using restrictions on

the domain and range, create the graphical representation of the left endpoint approximations on the interval [0, 6] for n = 3 and n = 12. You should create two separate graphs, one for each value of n. Shade the regions bounded by the rectangles using a color of your choice. The rectangles should be created using Demos.com, and not drawn in by hand.

3. (4 pts) Find the exact value for the area bounded by f(x) and the x-axis on the interval [0,6] using the LIMIT DEFINITION (this required the limit of a Riemann sum, using sigma notation appropriately). Show your work.

4. (1 pt) Based on the approximations (both numerical and graphical), do the approximations get better or worse as the value of n increases?

5. (1 pt) Calculate the exact error for each value of n, rounded to 2 decimal places if necessary (I am asking you to take your exact answer from #3 and subtract the approximations you found on the second and third page of the project).

• n = 3 error

• n = 6 error

• n = 12 error

Part 2: Trapezoidal and Simpson’s Rule

Use the integral ∫ 2

(𝑥−4)2 3

1 𝑑𝑥 to answer the following questions.

1. (2 pts) Estimate the definite integral using the Trapezoidal Rule with n = 4. Use

subintervals of equal width. Show your setup, but you can use a calculator for the tedious calculations. Round to 5 decimal places if necessary.

Estimate Answer:_______________________

2. (2 pt) Sketch a graph of f(x) = 2

(𝑥−4) 2, as well the trapezoids used for the approximation

on [1,3].

3. (2 pt) Evaluate the integral ∫ 2

(𝑥−4)2 3

1 𝑑𝑥 by hand using the Fundamental Theorem of

Calculus.

Use the integral ∫ 2

(𝑥−4)2 3

1 𝑑𝑥 to answer the following questions.

4. (2 pts) Estimate the definite integral using Simpson’s Rule with n = 4. Use subintervals

of equal width. Show your setup, but you can use a calculator for the tedious calculations. Round to 5 decimal places if necessary.

Estimate Answer:_______________________

Part 3: Introduction to Excel Open a blank workbook in Microsoft Excel (by going to start, all programs, Microsoft office, Microsoft Excel, and then left click the mouse). Type your first name in cell A1 and your last name in cell B1. Adjust the column widths as needed by moving the mouse to the top row of the page and placing the cursor on the right boundary of the cell you wish to adjust (cursor will appear as a plus sign or cross) and left click on the column and drag to widen. You can also change the column widths by highlighting the columns and clicking on Format, Column Width. Save your work to your computer or jump drive by clicking on FILE and then SAVE AS. Save the file as: LastNameFirstNameMath2231FA16 (Example: SmithJohnMath2231FA16). The following steps are designed to help you understand how formulas work in Excel: Move the cell pointer using the mouse. Left click over the desired cell or use the arrow keys to navigate the spreadsheet. In cell B4, type = (2 + 4)^2 + 3 ∗ 𝑠𝑞𝑟𝑡(4) then press Enter. What do you see in B4? Now in cell B5 type (2 + 4)^2 + 3 ∗ 𝑠𝑞𝑟𝑡(4) then press Enter. What do you see now? Move to cell B6 and type = 𝐵4 and the move to cell B7 and type 𝐵4. What is the difference? In cell A4 type 1 and in cell B4 edit the formula by replacing the number 4 in the square root function with 𝐴4. What is the value produced by cell A4? Type different values in cell A4 and notice the changes in outputs in B4. What happens when you enter −1 in cell A4? Now delete all cells besides A1 and B1 before beginning the project. You can click and drag the cursor to highlight several cells at once and then press Delete.

Part 4: Excel Document Setup • Open a blank workbook in Microsoft Excel

• In cell A1, type your first name

• In cell B1, type your last name

• In cell C1, type our course name (Math 2231)

• In cell E1, type Left Endpoint Approximations

• In cell A2, type Value of i

• At the bottom left hand side of the page, left click on the plus symbol to add a new page. Repeat this process until you have 5 pages total. Rename the pages by right clicking on the tab, then select rename. Name the first tab “Left Endpoints,” the second tab “Right Endpoints,” the third tab “Midpoints,” the fourth tab “Trapezoidal Rule,” and the fifth tab “Simpson’s Rule”

Part 5: Comparison of Approximation Techniques We will be calculating the left endpoint, midpoint, right endpoint, Trapezoidal Rule, and

Simpson’s Rule approximations for the function from part 1: f(x) = 𝑥3

3 −

7𝑥2

3 +

10𝑥

3 + 4.

For the “Left Endpoints” Tab: Column A: Value of i

• In cell A3, type = 1 (the starting point for our index)

• In cell A4, type = 𝐴3 + 1

• Now copy cell A4 (ctrl c). Click and drag your cursor to highlight cells A5 through A1003, then paste your copied cell (ctrl v).

Column B: Left Endpoints for various values of n

• In cell B2, type 𝑛 = 4 (use bold font)

• In cell B5, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 4

• In cell B6, type = (𝐴3 − 1) ∗ 6/4

• Now copy cell B6 (ctrl c). Click and drag your cursor to highlight cells B7 through B9, then paste your copied cell (ctrl v).

• In cell B12, type 𝑛 = 6 (use bold font)

• In cell B15, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 6

• In cell B16, type = (𝐴3 − 1) ∗ 6/6

• Now copy cell B16 (ctrl c). Click and drag your cursor to highlight cells B17 through B21, then paste your copied cell (ctrl v).

• In cell B24, type 𝑛 = 12 (use bold font)

• In cell B27, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 12

• In cell B28, type = (𝐴3 − 1) ∗ 6/12

• Now copy cell B28 (ctrl c). Click and drag your cursor to highlight cells B29 through B39, then paste your copied cell (ctrl v).

• In cell B42, type 𝑛 = 24 (use bold font)

• In cell B45, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 24

• In cell B46, type = (𝐴3 − 1) ∗ 6/24

• Now copy cell B46 (ctrl c). Click and drag your cursor to highlight cells B47 through B69, then paste your copied cell (ctrl v).

• In cell B72, type 𝑛 = 50 (use bold font)

• In cell B75, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 50

• In cell B76, type = (𝐴3 − 1) ∗ 6/50

• Now copy cell B76 (ctrl c). Click and drag your cursor to highlight cells B77 through B125, then paste your copied cell (ctrl v).

• In cell B128, type 𝑛 = 100 (use bold font)

• In cell B131, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 100

• In cell B132, type = (𝐴3 − 1) ∗ 6/100

• Now copy cell B132 (ctrl c). Click and drag your cursor to highlight cells B133 through B231, then paste your copied cell (ctrl v).

• In cell B234, type 𝑛 = 200 (use bold font)

• In cell B237, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 200

• In cell B238, type = (𝐴3 − 1) ∗ 6/200

• Now copy cell B238 (ctrl c). Click and drag your cursor to highlight cells B239 through B437, then paste your copied cell (ctrl v).

• In cell B440, type 𝑛 = 500 (use bold font)

• In cell B443, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 500

• In cell B444, type = (𝐴3 − 1) ∗ 6/500

• Now copy cell B444 (ctrl c). Click and drag your cursor to highlight cells B445 through B943, then paste your copied cell (ctrl v).

• In cell B946, type 𝑛 = 1000 (use bold font)

• In cell B949, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑓𝑜𝑟 𝑛 = 1000

• In cell B950, type = (𝐴3 − 1) ∗ 6/1000

• Now copy cell B950 (ctrl c). Click and drag your cursor to highlight cells B951 through B1949, then paste your copied cell (ctrl v).

Column C: Function Values at Left Endpoints: Height of Rectangle

• In cell C5, type 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑡 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠: 𝐻𝑒𝑖𝑔ℎ𝑡

• Copy cell C5 (ctrl c). Now paste (ctrl v) the cell into C15, C27, C45, C75, C131, C237, C443, and C949

• In cell C6, type = ((𝐵6)^3)/3 – (7/3) ∗ (𝐵6)^2 + (10/3) ∗ (𝐵6) + 4

• Now copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C7 through C9, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C16 through C21, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C28 through C39, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C46 through C69, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C76 through C125, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C132 through C231, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C238 through C437, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C444 through C943, then paste your copied cell (ctrl v).

• Copy cell C6 (ctrl c). Click and drag your cursor to highlight cells C950 through C1949, then paste your copied cell (ctrl v).

Column D: Individual Rectangle Sums: Rectangle Height x Width

• In cell D2, type 𝑊𝑖𝑑𝑡ℎ 𝑜𝑓 𝑠𝑢𝑏𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠 (𝑏 − 𝑎)/𝑛

• Copy cell D2 (ctrl c). Now paste (ctrl v) the cell into D12, D24, D42, D72, D128, D234, D440, and D946

• In cell D3, calculate the width of the subinterval for n = 4, and type in the number (note, if excel changes the cell to a date such as Jan 5, you can right click on the cell, select format cells, then on the left hand side select number. Make sure the decimal places box shows at least 4. If any other boxes change to dates, follow this same process to make the cell a number).

• In cell D5, type 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚𝑠: 𝐻𝑒𝑖𝑔ℎ𝑡 𝑥 𝑊𝑖𝑑𝑡ℎ

• Copy cell D5 (ctrl c). Now paste (ctrl v) the cell into D15, D27, D45, D75, D131, D237, D443, and D949

• In cell D6, you will need to use the width of the subinterval for n = 4, let’s call the width of the subinterval w. Type = 𝐶6 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D6 (ctrl c). Click and drag your cursor to highlight cells D7 through D9, then paste your copied cell (ctrl v).

• In cell D13, calculate the width of the subinterval for n = 6, and type in the number.

• In cell D16, you will need to use the width of the subinterval for n = 6, let’s call the width of the subinterval w. Type = 𝐶16 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D16 (ctrl c). Click and drag your cursor to highlight cells D17 through D21, then paste your copied cell (ctrl v).

• In cell D25, calculate the width of the subinterval for n = 12, and type in the number.

• In cell D28, you will need to use the width of the subinterval for n = 12, let’s call the width of the subinterval w. Type = 𝐶28 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D28 (ctrl c). Click and drag your cursor to highlight cells D29 through D39, then paste your copied cell (ctrl v).

• In cell D43, calculate the width of the subinterval for n = 24, and type in the number.

• In cell D46, you will need to use the width of the subinterval for n = 24, let’s call the width of the subinterval w. Type = 𝐶46 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D46 (ctrl c). Click and drag your cursor to highlight cells D47 through D69, then paste your copied cell (ctrl v).

• In cell D73, calculate the width of the subinterval for n = 50, and type in the number.

• In cell D76, you will need to use the width of the subinterval for n = 50, let’s call the width of the subinterval w. Type = 𝐶76 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D76 (ctrl c). Click and drag your cursor to highlight cells D77 through D125, then paste your copied cell (ctrl v).

• In cell D129, calculate the width of the subinterval for n = 100, and type in the number.

• In cell D132, you will need to use the width of the subinterval for n = 100, let’s call the width of the subinterval w. Type = 𝐶132 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D132 (ctrl c). Click and drag your cursor to highlight cells D133 through D231, then paste your copied cell (ctrl v).

• In cell D235, calculate the width of the subinterval for n = 200, and type in the number.

• In cell D238, you will need to use the width of the subinterval for n = 200, let’s call the width of the subinterval w. Type = 𝐶238 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D238 (ctrl c). Click and drag your cursor to highlight cells D239 through D437, then paste your copied cell (ctrl v).

• In cell D441, calculate the width of the subinterval for n = 500, and type in the number.

• In cell D444, you will need to use the width of the subinterval for n = 500, let’s call the width of the subinterval w. Type = 𝐶444 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D444 (ctrl c). Click and drag your cursor to highlight cells D445 through D943, then paste your copied cell (ctrl v).

• In cell D947, calculate the width of the subinterval for n = 1000, and type in the number.

• In cell D950, you will need to use the width of the subinterval for n = 1000, let’s call the width of the subinterval w. Type = 𝐶950 ∗ 𝑤 (note: you need to replace w with the actual value you found).

• Copy cell D950 (ctrl c). Click and drag your cursor to highlight cells D951 through D1949, then paste your copied cell (ctrl v).

Column E: Left Endpoint Approximations

• In cell E1, type 𝐿𝑒𝑓𝑡 𝐸𝑛𝑑𝑝𝑜𝑖𝑛𝑡 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛𝑠

• In cell E5, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 4

• In cell E6, type = 𝑠𝑢𝑚(𝐷6: 𝐷9)

• In cell E15, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 6

• In cell E16, type = 𝑠𝑢𝑚(𝐷16: 𝐷21)

• In cell E27, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 12

• In cell E28, type = 𝑠𝑢𝑚(𝐷28: 𝐷39)

• In cell E45, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 24

• In cell E46, type = 𝑠𝑢𝑚(𝐷46: 𝐷69)

• In cell E75, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 50

• In cell E76, type = 𝑠𝑢𝑚(𝐷76: 𝐷125)

• In cell E131, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 100

• In cell E132, type = 𝑠𝑢𝑚(𝐷132: 𝐷231)

• In cell E237, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 200

• In cell E238, type = 𝑠𝑢𝑚(𝐷238: 𝐷437)

• In cell E443, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 500

• In cell E444, type = 𝑠𝑢𝑚(𝐷444: 𝐷943)

• In cell E949, type 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛 = 1000

• In cell E950, type = 𝑠𝑢𝑚(𝐷950: 𝐷1949)

For the “Right Endpoints” Tab: Repeat the same setup for this tab that you completed for the left endpoints. Make the necessary changes for the right endpoints. Complete the approximations for n = 4, n = 6, n = 12, n = 24, n = 50, n = 100, n = 200, n = 500, and n = 1000. It might be easiest to copy and paste the entire first tab into the right endpoints tab, make a change in one cell, and then copy and paste to the other cells as necessary. Be sure to change any headings from “left endpoints” to “right endpoints.”

For the “Midpoints” Tab: Repeat the same setup for this tab that you completed for the left and right endpoints. Make the necessary changes for the midpoints. Complete the approximations for n = 4, n = 6, n = 12, n = 24, n = 50, n = 100, n = 200, n = 500, and n = 1000. It might be easiest to copy and paste the entire first tab into the midpoints tab, make a change in one cell, and then copy and paste to the other cells as necessary. Be sure to change any headings from “left endpoints” to “midpoints.”

For the “Trapezoid Rule” Tab: We will be completing the Trapezoidal Rule for n = 4, n = 6, n = 12, n = 24, n = 50, and n = 100 Column A:

• On the “Left Endpoints” tab, use your cursor to highlight cells A1 through A232. Copy these cells, and then select the “Trapezoidal Rule” tab. Click on cell A1, then paste the copied cells.

Column B:

• On the “Left Endpoints” tab, use your cursor to highlight cells B1 through B232. Copy these cells, and then select the “Trapezoidal Rule” tab. Click on cell B1, then paste the copied cells. In column B, change all the headings that say “Left Endpoints for n = ” to say “Interval Endpoints for n = ”

• Copy cell B9, and paste this copied cell into B10.

• Copy cell B21, and paste this copied cell into B22

• Copy cell B39, and paste this copied cell into B40

• Copy cell B69, and paste this copied cell into B70

• Copy cell B125, and paste this copied cell into B126

• Copy cell B231, and paste this copied cell into B232 Column C

• On the “Left Endpoints” tab, use your cursor to highlight cells C1 through C232. Copy these cells, and then select the “Trapezoidal Rule” tab. Click on cell C1, then paste the copied cells.

• Change all the headings that say “Function values at left endpoints: Height” to say “Function values for the Trapezoidal Rule (times 2 for the middle values)”

• Copy Cell C9, and past this copied cell into C10

• Copy Cell C21, and past this copied cell into C22

• Copy Cell C39, and past this copied cell into C40

• Copy Cell C69, and past this copied cell into C70

• Copy Cell C125, and past this copied cell into C126

• Copy Cell C231, and past this copied cell into C232

• For the Trapezoidal Rule, each middle function value needs to be multiplied by 2. Thus all cells in column C that represent function values, except the first and last function value on the interval, needs to be multiplied by 2. For example, the formula in cell C7 needs to change from = ((𝐵7)^3)/3 − (7/3) ∗ (𝐵7)^2 + (10/3) ∗ (𝐵7) + 4 into = 2 ∗ (((𝐵7)^3)/3 − (7/3) ∗ (𝐵7)^2 + (10/3) ∗ (𝐵7) + 4) (be very careful with your parenthesis). Make this change for n = 4, n = 6, n = 12, n = 24, n = 50, and n = 100

Column D

• Type “Sum of Function Values for the Trapezoidal Rule” in cells D5, D15, D27, D45, D75, D131

• In cell D6, type = 𝑠𝑢𝑚(𝐶6: 𝐶10)

• In cell D16, type = 𝑠𝑢𝑚(𝐶16: 𝐶22)

• In cell D28, type = 𝑠𝑢𝑚(𝐶28: 𝐶40)

• In cell D46, type = 𝑠𝑢𝑚(𝐶46: 𝐶70)

• In cell D76, type = 𝑠𝑢𝑚(𝐶76: 𝐶126)

• In cell D132, type = 𝑠𝑢𝑚(𝐶132: 𝐶232) Column E

• In cell E1, type Trapezoidal Approximations

• In cell E5, type “Sum times (b - a)/2n Approximation for n = 4”. For the following approximations, follow the same pattern to label the headings appropriately. These headings should occur in cell E15, E27, E45, E75, and E131.

• We will be multiplying the sum found in column D times (b - a)/2n for the various values of n. You will need to calculate the value for (b - a)/2n for n = 4, n = 6, n = 12, n = 24, n = 50, and n = 100 to complete the Trapezoidal Rule approximations. Then in cell E6, E16, E28, E46, E76, and E132 you need to multiply the sum found in column D times the value (b - a)/2n. Be sure to put = before the multiplication, and be sure the numbers you get for the approximations seem reasonable.

For the “Simpson Rule” Tab: We will be completing Simpson’s Rule for n = 4 and n = 6. Use the same process we did for the Trapezoidal Rule to create formulas for Simpson’s Rule. It would probably be easiest to copy the Trapezoidal Rule tab for n = 4 and n = 6 and paste into the Simpson’s Rule tab. Change the necessary headings to read “Simpson’s Rule” instead of “Trapezoidal Rule.” You will need to change columns C and E. In column C, Simpson’s Rule requires multiplying by 2 or 4 for alternating function values. Look at the formula for Simpson’s Rule to input the correct numbers into Excel. In column E, you will need to multiply your sum by (b - a)/3n instead of (b - a)/2n. Be sure your approximations seem reasonable.