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Math IB SL Shady Areas Portfolio

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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 image00.pngimage00.pngtrapezoids. 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: image21.pngimage21.png is given:

image96.png

Figure 1 – Graph of image102.pngimage102.png

Using two trapezoids mapped onto the curve in the domain image114.pngimage114.png, 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).

...read more.

Middle

image114.png:

image121.png

Diagram 3 – Illustration of how to map image122.pngimage122.pngtrapezoids onto image120.pngimage120.png

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 image93.pngimage93.png  for this specific case. image94.pngimage94.png can be replaced with image123.pngimage123.png, where n is the number of trapezoids used.  a+b can be replaced with: image124.pngimage124.png for the first trapezoid, image02.pngimage02.png for the second trapezoid, image03.pngimage03.pngfor the third and so on. Because image04.pngimage04.png is 1, the second to last a+b replacement will be image05.pngimage05.png, and the last one image06.pngimage06.png

This gives the following formulas for the area of each trapezoid:

image07.pngimage07.png

image09.pngimage09.png

image10.pngimage10.png

image11.pngimage11.png

image12.pngimage12.png

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:

image13.png

image14.pngimage14.png

Which can be factored by taking out the values of  image15.pngimage15.png:

image16.pngimage16.png

Noticing the terms image17.pngimage17.png – where image18.pngimage18.png is any whole number – each appear twice, the expression can be further simplified:

image20.pngimage20.png

The above is now the general expression for finding the area under the curve in the function image21.pngimage21.png, from image22.pngimage22.png to image23.pngimage23.png using image00.pngimage00.pngtrapezoids.

In order to develop from that a general statement that will estimate the area under any curve where image24.pngimage24.pngand in the general domain ofimage25.pngimage25.png using n trapezoids, one can modify the formula to accept the variables image26.pngimage26.png instead of image27.pngimage27.png andimage28.pngimage28.png instead of image30.pngimage30.png, as well as replacing image31.pngimage31.png with image32.pngimage32.pnggiving:

image33.png

...read more.

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:

 Shady Areas

By Calvin Ho

November 24th, 2009

...read more.

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

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