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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

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

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 function  The diagram below shows the graph of g. The area under this curve from to Middle

Now, I will increase the number of trapeziums to five and find a second approximation for the area.  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.  8 Trapeziums  9 Trapeziums  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. So basically, all I did was use the trapezoid rule as my base and just replace the   with 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 from to using n

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

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