﻿ GPC - John Weber - Math2431 Computer Project 5 - 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.

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

(* 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 xi using `xi = a + i*dx`
Evaluate f(xi) for all xi 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 xi using `xi = a + i*dx`
Evaluate f(xi) for all xi 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 xi using `xi = a + (i - 1/2)*dx`
Evaluate f(xi) for all xi 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`.

### 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.