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.

In this project, you will use explore how Reimann Sums estimate the area between the curve,
f(x) = 2x^{2} – x^{3}/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.

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 *)

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 R_{4} using `R4 = Total[f[xi]*dx]`

Draw the rectangles on your plot.

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 L_{4} using `L4 = Total[f[xi]*dx]`

Draw the rectangles on your plot.

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 M_{4} using `M4 = Total[f[xi]*dx]`

Draw the rectangles on your plot.

Plot the function using `Plot[f[x], {x, 0, 4}]`

Evaluate T_{4} using `T4 = (R4 + L4)/2`

Draw the trapezoids on your plot.

Evaluate S_{4} using `S4 = (2*M4 + T4)/3`

.

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

.

- Which Riemann sums (excluding the Simpson sum) are the best estimates? Explain.
- Which Riemann sum(s) (excluding the Simpson sum) are overestimate(s)? Explain.
- Which Riemann sum(s) (excluding the Simpson sum) are underestimate(s)? Explain.
- 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.