Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities: If, nd re rel numers, then: If < then ± < ± < for positive > for negtive Notie tht the diretion of the inequlity hnges when you multiply (or divide) y negtive numer For,,, d positive if nd d d then Reiprol reltions for nd positive. If > then > then < Informlly: As inreses dereses nd vie vers. Frequently n inequlity is of interest only within ertin intervl. For x, x + x x +. Mdison, Wisonsin Pge of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 2 of Asolute Vlues: For positive mens extly the sme thing s https://www.desmos.om/lultor/origddm Turn the vrious grphs on nd off. Nottion: f (x) x def x : x x : x < https://www.desmos.om/lultor/extvwgp9 The nottion : y(t) g(t) defines two funtion rules y(t) g(t) y (t) g(t) https://www.desmos.om/lultor/pvvw9yx Mdison, Wisonsin Pge 2 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Limits: Rememer tht the symols DNE ( ) lternte nottion ( ) ( DNE ) lternte nottion ( ) do not represent numers. Eh one represents non-terminting proess. Something is inresing or deresing without ound ut never getting lose to (pprohing) ny well-defined numer. They indite tht it Does Not Exist. See Slrs 2., setion on when the it Does Not Exist. In the following sttements P is onstnt. The vlues of nd/or re hnging. You ll e using these results lot inside more omplited its so hve them t your fingertips. Memorize the first three results in olumn. P > P P < P DNE ( ) P P DNE P ( ) or ( DNE ) (depends on P) P > P P < P P P P P ( DNE ) ( DNE ) or ( DNE ) (depends on P) Mdison, Wisonsin Pge of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 4 of The Min Topi - Improper Definite Integrls The definition of the definite integrl f (x)dx tht we hve used so fr requires tht the its of integrtion nd e finite nd f (x) e ontinuous within the losed intervl [, ]. We now extend tht definition to inlude the following. DEFINITIONS: Improper Definite Integrls. Note tht the following re definitions. Type I: Non finite its of integrtion ( nd ). Let f (x) F (x). Whenever the its elow exist: For ny hoie of : You n t do: f (x)dx ( ( F() F() ) ) Type II: Finite its of integrtion ut f (x) inreses or dereses without ound for t lest one point in the rnge of integrtion (n e n endpoint or n interior point). Alwys exmine f (x) to determine if this hppens. If you do not hndle this kind of prolem properly then you my get the wrong nswer. DEFINITION: Type II Improper Integrls. f (x)dx def x f (x)dx x F(x ) F() x [ ] f ontinuous x <, prolem t x f (x)dx def x + f (x)dx def f (x)dx + x f (x)dx f (x)dx x + [ F() F(x )] f ontinuous < x, prolem t x f ontinuous x, exept prolem t x where < < There re three possile vlues for eh of the integrls ove.. Converges to it (hs finite vlue).. Diverges DNE osilltes without onverging to it. See Slrs 2., the setion on when the it Does Not Exist (DNE). Mdison, Wisonsin def f (x) dx Pge 4 of It is interesting tht we hve the it of it: n n k f (x + kδx)δx
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 5 of ( ) ( ) or ( DNE ) ( ). Diverges. DNE The following exmples demonstrte tht the shpe of grph does not tell you if definite integrl onverges. Py speil ttention to the vlue of p for integrls of the form or or. This will e importnt lter. EXAMPLE: The Definite Integrl DNE. dx x 2 x dx x 2 ( 2 ) ( DNE ) https://www.desmos.om/lultor/uenvku5gyn 2 2 2 2 ( ) DNE ( ) 2 EXAMPLE: The Definite Integrl DNE. x dx def x dx ln(x) ( DNE ) https://www.desmos.om/lultor/imy6zyus ( ) ln ln() ( DNE ) However: EXAMPLE: The Definite Integrl onverges. dx def x 2 x x 2 dx x + https://www.desmos.om/lultor/jjhhnk5jxm Mdison, Wisonsin Pge 5 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 6 of Here re the sme integrls using x insted of x. EXAMPLE: The Definite Integrl DNE dx ( x) 2 dx ( x) 2 x ( 2 ) 2( x) /2 2 ( ) /2 2( ) /2 2 2 DNE ( ) /2 ( ) 2 + 2 DNE ( DNE ) https://www.desmos.om/lultor/dxld5l4hr where is negtive EXAMPLE: The Definite Integrl DNE. x dx def x dx ln x ( ln ) ln [ ] ( ) ( ) ( ) ln DNE ln DNE ( DNE ) https://www.desmos.om/lultor/eypsseqe EXAMPLE: The Definite Integrl onverges (to finite vlue). dx def x 2 x x 2 dx x + https://www.desmos.om/lultor/vfwv6v9r Mdison, Wisonsin Pge 6 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 7 of Here is the more generl se. EXAMPLE: For p dx def x p x p dx p p p p ( p ) p p When p is positive For p > nd p ( p ) ( ) > For p < nd ( p ) ( p ) ( DNE ) For p > For p < For p dx dx p ( ) p DNE p Converges ( ) DNE Diverges dx Derived erlier DNE Diverges x Cross hek: In words: For > nd dx the rekpoint for onvergene is p. If p is ever so slightly greter thn dx, x. integrl onverges. Otherwise it diverges. dx, x 2 dx x then the definite Mdison, Wisonsin Pge 7 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 8 of EXAMPLE: The definite integrl DNE (Does Not Exist). os x dx DNE The vlue of the integrl osilltes s x grows without ound. It does not onverge to it. https://www.desmos.om/lultor/dxsqjuturg EXAMPLE: TYPE I For > e x dx def e x dx e x e e + So, this integrl lwys onverges to it. https://www.desmos.om/lultor/ydhqfvv Mdison, Wisonsin Pge 8 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge 9 of Here is list of definite integrls tht re frequently used s the omprison integrl g(x) in the theorems of the next hpter. EXAMPLE: TYPE II f (x) inreses without ound t n endpoint. dx Oserve tht x 2 x dx x dx x 2 dx DNE DNE dx DNE < p dx e x dx p def p > > x 2 is not defined t x. x dx rsin x 2 So it's improper nd we use the definitinon ( ) rsin() rsin() π 2 π 2 [ rsin() rsin() ] The vlue of the integrl is the it of the re of the shded region elow s pprohes. https://www.desmos.om/lultor/jzqd5v4 Mdison, Wisonsin Pge 9 of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of TYPE II f (x) inreses with ound t n interior point. Thoms th. Ed. p. 624. EXAMPLE: https://www.desmos.om/lultor/qvittexdu ghv f (x) inreses without ound t x so we use the Type II definition. Now we onsider two different integrls. dx ( x ) 2/ dx + ( x ) 2/ dx x ( ) 2/ dx + 2 x ( ) 2/ ( x ) 2/ dx x ( ) 2/ x dx ( ) / ( ) / ( ) / ( ) / ( ) / + ( x ) 2/ dx x + ( ) 2/ x dx + ( ) / ( ) / ( ) / ( 2) / ( ) / ( 2) / + 2 Mdison, Wisonsin Pge of
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Review of Types of Improper Integrls Type I. Non Finite Limits of Integrtion Type II. Finite Limits of Integrtion f (x) inreses or dereses without ound Mdison, Wisonsin Pge of