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Unformatted text preview: CHAPTER 2 SOME APPLICATIONS OF INTEGRATION Chapter 2 is a bunch of somewhat unrelated topics, each of which illustrates an example of how you can give a meaning to an integral. Note I have only written down what I think are the most important definitions and theorems in here. I expect you to know (roughly) how the proofs work, and you should read those in the book. Moreover I will not restrict myself to ask just about these items on the midterm/final; the full subject of those exams will be determined by a list of sections in the book. 1. Physical interpretations If f ( x ) describes the rate of change of some physical system, then R b a f ( x ) dx describes how much the system has changed if you moved from a to b . For example if f ( x ) is your speed as a function of time, then R f ( x ) dx gives the distance travelled (during a given period of time). Also if f ( x ) describes the change in elevation per distance (for a road), then R f ( x ) dx (integral over distance) gives the total difference in height over the entire trip. In the book this is illustrated by the concept of work in Sections 2.14-2.15 2. Areas Areas are discussed in Sections 2.2-2.4. We already know that the integral of a positive function is the area of its ordinate set. Slightly generalizing we immediately find Theorem 2.1. Let f and g be integrable functions, and let f ≤ g on [ a,b ] then the area of the region S = { ( x,y ) | x ∈ [ a,b ] ,f ( x ) ≤ y ≤ g ( x ) } is given by Z b a g ( x )- f ( x ) dx. As an important example we find that the area of a disc of radius r is given by Z r- r p r 2- x 2 dx = r 2 Z- 1 1 p 1- x 2 dx =: πr 2 , where the last equation defines π for us....
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This note was uploaded on 11/19/2010 for the course ANTH 122 taught by Professor 323 during the Spring '10 term at Centennial College.

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