This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 2.142.15 2. Areas Areas are discussed in Sections 2.22.4. 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....
View
Full
Document
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.
 Spring '10
 323

Click to edit the document details