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Integral Portfolio

Extracts from this document...

Introduction

Trapezium Portfolio

Raghav “Ron” Nair

Scot Millhollen

IB/AP Calculus SL

January 21, 2009


Table Of Contents

Introduction Page…………………………………………………………………………        1

Table of Contents…………………………………………………………………………. 2

Goal……………………………………………………………………………………………..        3

First Function and Five Trapeziums………………………………………………        4

Seven Trapeziums………………………………………………………………………..        5

Eight Trapeziums………………………………………………………………………….        6

Nine Trapeziums…………………………………………………………………………..        7

General Expression with “n” Trapeziums………………………………………        8

The General Statement…………………………………………………………………        9

Three New Curves…………………………………………………………………………        10                

Curve # 1……………………………………………………………………………………….        11

Curve # 2……………………………………………………………………………………….        12

Curve # 3……………………………………………………………………………………….        13

Actual Area of the Curves………………………………………………………………        14

Actual Area vs. Approximation……………………………………………………… 15

Trapezium Rule vs. Riemann Sums………………………………………………..        16

Scope and Limitations……………………………………………………………………        17

Disclaimer…………………………………………………………………………………….        18

Intentionally Blank


Goal

In this investigation, I will attempt to find a rule to approximate the area under a curve (i.e. between the curve and the x-axis) using trapeziums (trapezoids). Also throughout this investigation, I will be using all sorts of technology varying from an online version of a TI-83 to online graphs.

First Function and 5      Trapeziums

To start, I will take a look at the functionimage00.pngimage01.png

The diagram below shows the graph of g. The area under this curve fromimage12.pngtoimage22.png

...read more.

Middle

Now, I will increase the number of trapeziums to five and find a second approximation for the area.

image40.pngimage00.png

7 Trapeziums

With the help of technology, I have created three diagrams showing an increasing number of trapeziums. Alongside each graph, I will also find the approximation for the area.

image41.png

image02.png


8 Trapeziums

image03.png

image04.png


9 Trapeziums

image05.png

image06.png

With the help of technology, my online version of a TI-83, I found the exact integral to be 3.333333. So, I noticed that the more trapezoids I use, the closer the approximation for the area under the curve.


General Expression

I will now find a general expression for the area under the curve of g, from x = 0 to x =1, using n trapeziums.

image07.png

So basically, all I did was use the trapezoid rule as my base and just replace the image08.pngimage09.pngimage10.pngwith actual numbers and n for the amount of trapeziums.


The General Statement

So to find the general statement what will estimate the area under any curve image11.pngfrom image13.pngto image14.pngusing n

...read more.

Conclusion

In conclusion, this method of finding the area under the curve is quite accurate at finding the area under the curve. It is more accurate than the upper, lower, and middle Riemann sums but it still less accurate than just mathematically calculating it with an integral.


Disclaimer:

This portfolio assignment represents my own work. I did not seek or receive any unauthorized assistance.

Name:                        Ron Nair                        

X_________________________________________

Date:                       January 21, 2009                        


Do Not Write

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Do Not Write!

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