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# Math IB SL Shady Areas Portfolio

Extracts from this document...

Introduction

This portfolio is an attempt at deriving and examining the scope and limitations of a general statement that can approximate the area under a curve using trapezoids. Generally, calculus – specifically the method of integration, is used to find the exact area under a curve. Although this method will be explored in comparison later in the portfolio, this investigation deals mainly with investigating a method to approximate the area using high school level math. First, this portfolio will attempt to derive a general statement that will give an approximation of the area under the curve of any function in any closed interval using  trapezoids. Then, by applying the formula to sample functions, the answers given can be compared to the integral answers, allowing an examination into the accuracy of the trapezoid method of approximation. Lastly, by examining different behaviors of a graph, this portfolio will investigate the cause of any inaccuracies in this method.

The graph:  is given: Figure 1 – Graph of  Using two trapezoids mapped onto the curve in the domain  , the area under the curve in that domain can be approximated as the sum of the areas of the two trapezoids.

In order to map the trapezoids onto the graph, one must first divide the domain by the number of trapezoids being used, in order to find the height of the trapezoids (which is equivalent to each other).

Middle : Diagram 3 – Illustration of how to map  trapezoids onto  To begin, one needs to find the area of each of the trapezoids that will be mapped onto this function. That will be done by manipulating the area of a trapezoid formula of  for this specific case.  can be replaced with  , where n is the number of trapezoids used.  a+b can be replaced with:  for the first trapezoid,  for the second trapezoid,  for the third and so on. Because  is 1, the second to last a+b replacement will be  , and the last one  This gives the following formulas for the area of each trapezoid:          In adding the areas of multiple trapezoids, the separate area formulas for each of them will be terms in a calculation whose sum gives total area, as was demonstrated in the calculation for 5 trapezoids. Thus:   Which can be factored by taking out the values of  :  Noticing the terms  – where  is any whole number – each appear twice, the expression can be further simplified:  The above is now the general expression for finding the area under the curve in the function  , from  to  using  trapezoids.

In order to develop from that a general statement that will estimate the area under any curve where  and in the general domain of  using n trapezoids, one can modify the formula to accept the variables  instead of  and  instead of  , as well as replacing  with  giving: Conclusion

Appendix:

Appleby, A., Letal, R., & Ranieri, G. (2007). Geometric Series: Sigma Notation. Pure Math 30 (pp. 177). Calgary: Absolute Value Publications.

Integral. (n.d.). Wikipedia, the free encyclopedia. Retrieved November 23, 2009, from http://en.wikipedia.org/wiki/Integral

Stewart, J., Davison, T. M., & Ferroni, B. (1989). Area: Numerical Methods. Calculus: A First Course (pp. 481). Toronto: McGraw-Hill Ryerson.

Trapezoid. (n.d.). Wikipedia, the free encyclopedia. Retrieved November 21, 2009, from http://en.wikipedia.org/wiki/Trapezoid

Trapezoidal Rule. (n.d.). Wikipedia, the free encyclopedia. Retrieved November 21, 2009, from http://en.wikipedia.org/wiki/Trapezoidal_rule

Technology Used:

Graphmatica Free Trial by kSoft

TI 84-Plus Silver Edition Graphing Calculator

Microsoft Office Word 2007

IB Math SL Type 1 Internal Assessment:

By Calvin Ho

November 24th, 2009

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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