... introducing method minimizes the algebra as well as the number of computations for certain types of integrals. It is important to note that we are not suggesting that the conventional methods for teaching techniques of integration be replaced with the innovation we are proposing. Rather, the new approach we suggest is to be presented only after the traditional techniques of integration by parts and substitution have been taught. The new approach is called the probability density function (p.d.f.) method and uses the properties of probability density functions to evaluate integrals for certain classes of functions. In this paper we demonstrate that, for certain classes of functions, this method greatly reduces the length and complexity of many integration problems. Key words: Probability Density Function (p.d.f.), Calculus, Integration Techniques, Gamma Function, Beta Function, Cauchy Function, Normal function

INTRODUCTION

Learning and practicing integration techniques is a standard requirement in the second semester calculus course. Unfortunately, some methods of integration can be difficult and time consuming for students, especially the method known as integration by parts. While introduced to calculus students as a means of obtaining a simpler, solvable integral from a more complex integral, integration by parts often involves multiple applications of the method in order to get the problem into a form simple enough to solve. The lengthy and tedious process involved in integration by parts often generates algebra errors that are hard to correct, leaving students frustrated with the method. Students may even find themselves asking if there is a better way to calculate some of the integrals requiring integration by parts. Fortunately, there is a better way. Some integrals can be evaluated using the properties of probability density functions for continuous random variables, thus simplifying and shortening the work.

A probability density function (p.d.f.) is a function defined on an interval (a, b) and having both of the following properties:

(i) f(x)[greater than or equal to]0 for every x, and

(ii) a.[integral].b f(x)dx = 1.

In (ii) it is important to note that a and/or b can be infinite, creating an improper integral. These properties of p.d.f.'s would likely be unfamiliar to the typical calculus student since p.d.f.'s are usually not taught in calculus courses. However, most statistics students are very familiar with the properties of p.d.f.'s.

One p.d.f. that is especially useful in calculus is the gamma function (3), which is given by f(x) = 1/[GAMMA]([alpha])[[beta].sup.[alpha]] [x.sup.[alpha]-1][e.sup.-x/[beta]]; 0, [beta]>0; and [GAMMA]([alpha]) = ([alpha]-1)! and is denoted by X~[GAMMA]([alpha], [beta]). Since the gamma function (3) is a p.d.f., it integrates to one for all values of x, giving [infinity].[integral].0 1/[GAMMA]([alpha])[[beta].sup.[alpha]] [x.sup.[alpha]-1][e.sup.-x/[beta]] dx = 1; where[alpha]>0, [beta]>0; and [GAMMA]([alpha]) = ([alpha]-1)!.

Calculus books contain numerous integrals of...

NOTE: All illustrations and photos have been removed from this article.



More articles from Georgia Journal of Science
The Georgia Academy of Science: affiliated with the American Associati..., June 22, 2006
Georgia Academy of Science., June 22, 2006

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