Approximate Integration

Transcription

1 Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio: The first situtio rises from the fct tht, i order to evlute f ( ) d usig the Fudmetl Theorem of Clculus (FTC), we eed to kow tiderivtive of f. However, sometimes, it is difficult, or eve impossile, to fid tiderivtive. For emple, it is impossile to evlute the followig itegrls ectly: e d + 3 d Situtio: The secod situtio rises whe the fuctio is determied from scietific eperimet through istrumet redigs or collected dt. There my e o formul for the fuctio (s we will see i Emple 5). I oth cses, we eed to fid pproimte vlues of defiite itegrls. Whe we eed to pproimte vlues of defiite itegrls, we hve to cosider: How fr from the truth c the pproimtio e? How much error is possile? How fst do the errors decrese s the umer of su divisios icreses? Recll: The defiite itegrl is defied s limit of Riem sums, so y Riem sum could e used s pproimtio to the itegrl. Left Edpoit Approimtio: f ( ) d L = f ( ) Δ If f(), the itegrl represets re d the equtio represets pproimtio of this re y the rectgles show here. The pproimtio L is clled the left edpoit pproimtio. i= i

2 Right Edpoit Approimtio: f ( ) d R = f ( ) Δ i= i The pproimtio R is clled right edpoit pproimtio. Midpoit Rule: The figure shows the midpoit pproimtio M. M ppers to e etter th either L or R. Trpezoidl Rule: The reso for the me c e see from the figure, which illustrtes the cse f().

3 Emple : Approimte the itegrl () Trpezoidl Rule (/ ) d with = 5, usig: () Midpoit Rule Actul Vlue: d = l ] = l = Let us fid the error of ech pproimtio: (Actul Vlue) (Approimtio) From the vlues i Emple, we see tht the errors i the Trpezoidl d Midpoit Rule pproimtios for = 5 re: E T.488 E M.39 I geerl, we hve: E = f ( ) d T T E = f ( ) d M M

4 The figure shows why we c usully epect the Midpoit Rule to e more ccurte th the Trpezoidl Rule. The re of typicl rectgle i the Midpoit Rule Is the sme s the re of Whose upper side is tget to the grph t p. The re of this trpezoid is closer to the re Uder the grph tht is the re of tht used i the. The midpoit error is smller th the trpezoidl error. Error Bouds Suppose f () K for. If E T d E M re the errors i the Trpezoidl d Midpoit Rules, the E T K ( ) ( ) d K E M Emple : Apply the error estimte to the Trpezoidl Rule Approimtio for the itegrl (/ ) d with =5.

5 Emple 3: Cosider the itegrl d. A. Use () Left Edpoit Approimtio, () Right Edpoit Rule, (c) Midpoit Rule, (d) Trpezoidl Rule with =5 to pproimte the itegrl d. () d L5 = () d R5 = (c) d M 5 = (d) d T5 = B. Which is etter pproimtio d more ccurte? C. Fid the error of ech pproimtio. ( ActulVlue Approimtio) () E T () E T

6 D. Cosider the followig pproimtios to d d correspodig errors. L R T M E L E R E T E M Oservtios:. By icresig the vlue of, we get. (We hve to ewre of ccumulted roud-off error.). The errors i L d R re. 3. T d M re much more ccurte th. 4. The error i T d M re. 5. The size of the error i is out hlf the size of the error i. 6. Errors of T d M roughly iversely proportiol to. 7. is etter th i every cse. 8.Errors of S re roughly iversely proportiol to. Notice tht Oservtio 4 correspods to the i ech deomitor ecuse: () = 4

7 E. Determie whether ech estimte is over or uderestimte.,, the ) If f is decresig fuctio d cocve up o [ ] f ( ) d ) If f is icresig fuctio d cocve up o [, ], the f ( ) d c) If f is decresig fuctio d cocve dow o [, ], the f ( ) d d) If f is icresig fuctio d cocve dow o [, ], the f ( ) d Simpso s Rule: S = f ( ) d S Δ = [ f ( ) + 4 f ( ) + f ( ) + 4 f ( 3) f ( ) + 4 f ( ) + f ( )] where is eve d = ( )/. Emple 4: Use Simpso s Rule with = to pproimte.

8 SIMPSON S RULE VS. MIDPOINT RULE The tle shows how Simpso s Rule compres with the Midpoit Rule for the itegrl, whose true vlue is out This tle shows how the error E s i Simpso s Rule decreses y fctor of out 6 whe is douled. Tht is cosistet with the pperce of 4 i the deomitor of the followig error estimte for Simpso s Rule. It is similr to the estimtes give i (3) for the Trpezoidl d Midpoit Rules. However, it uses the fourth derivtive of f. ERROR BOUND (SIMPSON S RULE) Suppose tht f (4) () K for. If E s is the error ivolved i usig Simpso s Rule, the Emple 5: How lrge should we tke to gurtee tht the Simpso s Rule pproimtio for is ccurte to withi.?

9 Let Δ =. Let =, = + Δ, = + Δ,, + Δ =. Left Edpoit Rule: f ( ) d L = Right Edpoit Rule: f ( ) d R = Midpoit Rule: f ( ) d M = Where _ + =, _ + =, _ 3 + _ 3 =,.., = +. Trpizoidl Rule: f ( ) d T = Simpso s Rule: f ( ) d S = Error Boud for Trpezoidl Rule Let f '' ( ) K for. E T = Error Boud for Midpoit Rule Let f '' ( ) K for. E M = Error Boud for Simpso s Rule Let f (4) ( ) K for. E S =

10 Sheet (7.8) Improper Itegrls I this sectio, we will ler: How to solve defiite itegrls where the itervl is ifiite d where the fuctio hs ifiite discotiuity. Improper Itegrls re clled improper for oe of two differet cses. Cse : The itervl is ifiite. Cse : f hs ifiite discotiuity i [, ] I either cse, the itegrl is clled improper itegrl. Cosider the ifiite regio S tht lies: Uder the curve y = / Aove the -is To the right of the lie = d = Fid: () d = () d = (c) d = Do you thik d =?

11 Let us evlute the itegrl from = to = t. The re of the prt of S tht lies to the left of the lie = t (shded) is: Notice tht A(t) < o mtter how lrge t is chose. We lso oserve tht: lim At ( ) = lim = t t t The re of the shded regio pproches s t. So, we sy tht the re of the ifiite regio S is equl to d we write: t d = lim d = t Ad we sy tht d to.

12 Cosider the itegrl d = d. Fid: () d = () d = (c) d = (c) d = Emple : Determie whether the itegrl d is coverget or diverget. So d is. () Steps for Evlutig Improper Itegr () (3)

13 Compre: d d d d d d P-Vlue d p d p

14 Improper Itegrl of Type : Defiitio (): t If f ( ) d eists for every umer t, the provided this limit eists (s fiite umer). Defiitio (): If t f ( ) d eists for every umer t, the provided this limit eists (s fiite umer). Coverget d Diverget The improper itegrls f ( ) d d f ( ) d re clled: if the correspodig limit eists. if the limit does ot eist. Defiitio (c): If oth f ( ) d d f ( ) d re coverget, the we defie: where y rel umer c e used. Ay of the improper itegrls i Defiitio c e iterpreted s re provided f is positive fuctio.

15 Emple : Evlute e d Emple 3: Evlute + It s coveiet to choose = d

16 Improper Itegrl of Type : Discotiuous Itegrds Suppose f is positive cotiuous fuctio defied o fiite itervl [, ) ut hs verticl symptote t. Let S e the uouded regio uder the grph of f d ove the -is etwee d. For Type itegrls, the regios eteded idefiitely i horizotl directio. Here, the regio is ifiite i verticl directio. The re of the prt of S etwee d t (shded regio) is: t A( t) = f ( ) d If it hppes tht A(t) pproches defiite umer A s t -, the we sy tht the re of the regio S is A d we write: f ()d = We use the equtio to defie improper itegrl of Type eve whe f is ot positive fuctio o mtter wht type of discotiuity f hs t. Defiitio 3 () : If f is cotiuous o [, ) d is discotiuous t, the if this limit eists (s fiite umer). Defiitio 3 (): If f is cotiuous o (, ] d is discotiuous t, the if this limit eists (s fiite umer). Defiitio 3 is illustrted for the cse where f() d hs verticl symptotes t d c, respectively.

17 The improper itegrl f ( ) d is clled: Coverget if the correspodig limit eists. Diverget if the limit does ot eist. Defiitio 3 (c): If f hs discotiuity t c, where < c <, d oth coverget, the we defie: c f ( ) d d c f ( ) d re Defiitio 3 c is illustrted for the cse where f() d hs verticl symptotes t d c, respectively. Emple 5: Fid 5 d Emple 6: Determie whether π sec d coverges or diverges.

18 Emple 7: Evlute 3 d if possile. WARNING: Suppose we hd ot oticed the symptote = 7 i Emple 7 d hd, isted, cofused the itegrl with ordiry itegrl. 3 3 The, we might hve mde the followig erroeous clcultio: d = l " # = l l This is wrog ecuse the itegrl is improper d must e clculted i terms of limits. = l From ow, wheever you meet the symol whether it is either: A ordiry defiite itegrl A improper itegrl f ( ) d, you must decide, y lookig t the fuctio f o [, ], Emple 8: Evlute l d

19 A Compriso Test for Improper Itegrls Sometimes, it is impossile to fid the ect vlue of improper itegrl d yet it is importt to kow whether it is coverget or diverget. I such cses, the followig theorem is useful. Although we stte it for Type itegrls, similr theorem is true for Type itegrls. Compriso Theorem: Suppose f d g re cotiuous fuctios with f() g() for.. If f ( ) d is coverget, the g ( ) d is.. If g ( ) d is diverget, the f ( ) d is. If the re uder the top curve y = f() is fiite, so is the re uder the ottom curve y = g(). If the re uder y = g() is ifiite, so is the re uder y = f(). The sic ide is: If you c pit regio, you c pit y regio iside (elow) this regio. IF you cot pit the regio, the you cot pit y regio outside (ove) the regio. Note tht the reverse is ot ecessrily true: If If g ( ) d is coverget, f ) d f ( ) d is diverget, g ) d ( my or my ot e coverget. ( my or my ot e diverget.

20 Emple 9: Show tht e d is coverget. Emple : The itegrl + e d is diverget y the Compriso Theorem sice + e > How to Apply the Compriso Test

21 WORKSHEET (7.8). Determie whether ech of the followig itegrls coverges or ot. 3 () d () d (c) 3 d (d) d 5 (e) d 4 (f) 4 3 d. Ivestigte the covergece of e 5 d.

22 3. Ivestigte the covergece of ( ) d. 4. Ivestigte the covergece of d. 4

23 5. Determie whether d coverges or diverges Determie whether d coverges or diverges. (l ) 4