- Level: International Baccalaureate
- Subject: Maths
- Word count: 2528
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 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:
Shady Areas
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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