MATH 2431 – Summer 2012

Project 5 – Riemann Sums – Due: Wednesday, 18 July 2012

You must complete the project in a group of two students. Normally, all members of the group will receive the same grade; however, the instructor reserves the right to conduct individual interviews over the content of the project and to assign different grades to different members of the group.

Introduction

In this project, you will use explore how Reimann Sums estimate the area between the curve, f(x) = 2x² – x³/4, and the x-axis on the interval, [0,4]. You will need to describe the relationship of the estimations. You will be graded on the quality and clarity of your written presentation as well as the mathematical accuracy of your paper.

Labeling Your Work

To label your work, use comments. For example, use the following to start your project (after every line, use SHIFT-ENTER to execute):

(* Your Names *)
(* Math2431-100 *)
(* Summer 2012 *)
(* Project 5 *)

1. The Right-Hand Sum, n = 4, i.e., R₄

Define the function using: f[x_] := 2 x^2 - x^3/4
Plot the function using Plot[f[x], {x, 0, 4}]
Store a [using a = 0], b [using b = 4], and n [using n = 4]
Store Δx into dx using dx = (b - a)/n
Find the index for the subintervals and store into i using i = Table[j, {j, 1, n}]
Find the values for x_i using xi = a + i*dx
Evaluate f(x_i) for all x_i using f[xi]
Find the areas of all the approximating rectangles using f[xi]*dx
Find R₄ using R4 = Total[f[xi]*dx]
Draw the rectangles on your plot.

2. The Left-Hand Sum, n = 4, i.e., L₄

Plot the function using Plot[f[x], {x, 0, 4}]
Find the index for the subintervals and store into i using i = Table[j, {j, 0, n-1}]
Find the values for x_i using xi = a + i*dx
Evaluate f(x_i) for all x_i using f[xi]
Find the areas of all the approximating rectangles using f[xi]*dx
Find L₄ using L4 = Total[f[xi]*dx]
Draw the rectangles on your plot.

3. The Midpoint Sum, n = 4, i.e., M₄

Plot the function using Plot[f[x], {x, 0, 4}]
Find the index for the subintervals and store into i using i = Table[j, {j, 1, n}]
Find the values for x_i using xi = a + (i - 1/2)*dx
Evaluate f(x_i) for all x_i using f[xi]
Find the areas of all the approximating rectangles using f[xi]*dx
Find M₄ using M4 = Total[f[xi]*dx]
Draw the rectangles on your plot.

4. The Trapezoid Sum, n = 4, i.e., T₄

Plot the function using Plot[f[x], {x, 0, 4}]
Evaluate T₄ using T4 = (R4 + L4)/2
Draw the trapezoids on your plot.

5. The Simpson Sum, n = 4, i.e., S₄

Evaluate S₄ using S4 = (2*M4 + T4)/3.

6. Repeat the above steps using n = 10 [you do not need to draw the rectangles/trapezoids].

7. Repeat the above steps using n = 20 [you do not need to draw the rectangles/trapezoids].

8. The Definite Integral

Use the following code: Integrate[f[x],{x,a,b}].

9. Questions

1. Which Riemann sums (excluding the Simpson sum) are the best estimates? Explain.
2. Which Riemann sum(s) (excluding the Simpson sum) are overestimate(s)? Explain.
3. Which Riemann sum(s) (excluding the Simpson sum) are underestimate(s)? Explain.
4. What happens to each sum as n gets larger? Explain.

Please follow these guidelines when preparing your report:

• Find a partner with whom you will complete the project. One person in the project needs to send the instructor an email in iCollege and cc: your partner. Type your partner's name and the project number on the Subject line of the email. This email is due on or before 10:00 a.m. on 11 July (1 point deducted if not completed).
• Each function must be graphed in order and clearly labeled.
• The entire project is to be submitted in class on or before 10:00 a.m. on the due date.
• The answers to the Questions must be typed (you may handwrite any unusual mathematical symbols, e.g., ∞).
• Use mathematically correct notation.
• Do not use any cover page or report cover.
• Turn in one paper per group. The instructor will keep all papers. Make a copy for your files before turning in your paper.
• Clearly explain your reasoning. Use complete, grammatically-correct sentences or complete mathematical sentences.