CALCULUSIICH15 - CH APTER 1 FURTH ER TECH NI QUES I N...

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Unformatted text preview: CH APTER 1 FURTH ER TECH NI QUES I N CONSTRUCTI NG ANTI DERI VATI VES CONTENTS 1.1. Review of Definite and I ndefinite I ntegr als 1.2. Tr igonometr ic I ntegr als 1.3. Par tial Fr actions 1.4. I ntegr ation Using Tables and Computer Algebr a Systems 1.5. I mpr oper I ntegr als. 1.5. I M PROPER I NTEGRALS 1.5.1. Type 1: I nfinite I nter vals 1.5.2. Type 2: Discontinuous I ntegr ands 1.5.3. Compar ison Test for I mpr oper I ntegr als 1.5.1 Type 1: I nfinite I nter vals Consider the area A of the infinite region S under the curve y =1/ x 2 , above the x- axis and on the right of x =1. At a first look, it seems that A is infinite. However if we look at the part of S on the left of the line x = t . then 1 1 1 1 1 ) ( 1 1 2 <- = - = = t x dx x t A t t 1 1 1 lim ) ( lim = - = t t A t t Moreover So we say that the area of the infinite region S is 1, and 1 1 lim 1 1 2 1 2 = = t t dx x dx x Definition of I mpr oper I ntegr al of Type 1. (a) If exists for all t a , then provided the limit exists (as a finite number) (b) If exists for all t b , then t a dx x f ) ( = t a t a dx x f dx x f ) ( lim ) ( b t dx x f ) ( - - = b t t b dx x f dx x f ) ( lim ) ( a dx x f ) ( - b dx x f ) ( We say that the improper integrals and are conver gent . Otherwise we say that they are diver gent Definition of I mpr oper I ntegr al of Type 1. (c) If both and are convergent, then we define the value on the right hand side does not depend on the choice of a - - + = a a dx x f dx x f dx x f ) ( ) ( ) ( a dx x f ) ( - a dx x f ) ( These improper integrals may be interpreted as areas of infinite regions if f ( x ) Example.Example....
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This note was uploaded on 11/02/2009 for the course ECE ece464 taught by Professor Abc during the Spring '09 term at Jacksonville College.

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CALCULUSIICH15 - CH APTER 1 FURTH ER TECH NI QUES I N...

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