Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist Cosmin NIŢU University of Agronomic Sciences nd Veterinry Medicine of Buchrest 59 Mărăşti Blvd District 464 Buchrest Romni Phone: +48564 F: + 48567Emil: lendru_n@yhoocom Astrct Corresponding uthor emil: nicosro@yhoocom Often in prctice one reches to inclculle integrls ut which cn e pproimted y numericl methods In fct in terms of ppliction there is no need for the ect result ut for knowing its vlue with n ccurcy no mtter how good In this pper we present two methods to pproimte Riemnn integrls: the method of rectngles nd trpezoids method After reviewing the theoreticl results we consider some pplictions focusing on the precision of pproimtions Key words: Riemnn integrl rectngle method trpezoidl method pproimtion precision INTRODUCTION Sometimes in our prctice work we otin integrls with incomputle primitives clled trnscendentl integrls In some cses these integrls cn e clculted with more dvnced techniques such s comple nlysis ut most of the times they cn only e pproimted y numericl methods The most elementry methods re linked to Riemnn sums: rectngles method trpezoids method Simpson s method Hermite s method Newton s method etc In this rticle we consider only first two methods We egin y proving the formuls nd then we pply them to severl integrls some of which hve importnt pplictions MATERIALS AND METHODS We follow (Grigore 99) (Muntenu nd Stnic 6) nd (Rosc ) for the proofs The rectngles method In mthemtics especilly in integrl clculus the rectngle method (lso clled the midpoint or mid-ordinte rule) computes n pproimtion to definite integrl mde y finding the re of collection of rectngles whose heights re determined y the vlues of the function Specificlly the intervl on which the function hs to e integrted is divided into n equl suintervls of length h n The rectngles re then drwn so tht either their left or right corners or the middle of their top line lies on the grph of the function with ses running long the O -is This process is illustrted y the net figures: Figure Figure Figure (see http://wwwmorg/eternl_rchive/jom/vol ume7/aktumen/rectnglehtml nd http://wwwmthcsemoryedu/~cheung/course s/7/syllus/7/rectngle-methodhtml) Inner Rectngles For the lower sum corresponding to inner rectngles we use Lefto pproimtion on the intervl [ ] nd the Righto pproimtion on the intervl [ ] Middle Rectngles To otin middle rectngles we simply use Middleo pproimtion on the entire intervl [ ] Outer Rectngles For the upper sum corresponding to outer rectngles we use Righto pproimtion on the intervl [ ] 99
nd the Lefto pproimtion on the intervl [ ] Figure The inner rectngle pproimtion The pproimtion to the integrl is then clculted y dding up the res (se multiplied y height) of the n rectngles giving the formul: n f ( d ) h f( ) i i h where n nd i ih The formul for i ove gives i for the Topleft corner pproimtion A grphic representtion of the rectngle method we cn follow in the net figures: Figure 4 The domin of studing the re of one function Figure The middle rectngle pproimtion The net step for us is to divide the specified re (the coloured one) in suintervls (rectngles) The more rectngles we hve the etter the pproimtion is For this spect the rectngles re then drwn so tht either their left or right corners or the middle of their top line lies on the grph of the function with ses running long the O -is: Figure The outer rectngle pproimtion Figure 5 Rectngles s suintervls
Figure 6 Dividing in suintervls For computing the entire intervl of studying we should summrize ll the suintervls we hve got As n gets lrger this pproimtion gets more ccurte In fct this computtion is the spirit of the definition of the Riemnn integrl nd the limit of this pproimtion s n is defined nd equl to the integrl of f on [ ] if this Riemnn integrl is defined Note tht this is true regrdless of which is used however the midpoint pproimtion tends to e more ccurte for finite n Figure 7 Computing the integrl function Numeric qudrture We re going to tke [ ] n distinct points of [] intervl ( division) nd f : R continuous function By qudrture formul we men n qulity of the type : n S f cif i i where ci R i n By numericl integrtion we men n pproimtion of the type: f ( d ) S( f) * For n h ( )/ n i ih i n i If S f is qudrture formul of the function f i i on the intervl i n then it cn e defined the summed qudrture formul: n n S f Si f i The rectngulr qudrture formul: The rectngulr qudrture formul is: S f P f d Considering Newton s formul of representing polynomil interpoltion we e get: P f f Then S( f) f d ( ) f f C If from the formul of evlution of the error t the interpoltion we will hve: ' f P f m f ( ) [ ] Then f d S f f P f d f P f d
' ' m f ( ) d m f ( ) [ ] S i f is rectngulr qudrture 4 [ ] formul pplied to the f function on the Oservtion: If f i i is derivle in the point intervl i n The trpezoidl method In numericl nlysis the trpezoidl rule (lso then the rectngulr qudrture known s the trpezoid rule or trpezium rule) formul cn e otined y the formul: is technique for pproimting the definite S f P f d integrl f C f ( d ) In tht cse if from the The trpezoidl rule works y pproimting formul of evlution of the error t the interpoltion we re going to hve: the region under the grph of the function f ( ) s trpezoid nd clculting its re It f P f follows tht f ( ) f( ) ' f( ) d ( ) m f ( ) [ ] Then: f ( d ) S( f) f( ) P f d f P f d ( ) m f ''( ) d m f ''( ) [ ] 4 [ ] The summed qudrture formul of the rectngle is: Figure 8 The trpezoidl rule n ( n) i i SD ( f) ( i i) f i Applicility nd lterntives n i i The trpezoidl rule is one of fmily of f n i formuls for numericl integrtion clled Newton Cotes formuls of which the midpoint n i i h f rule is similr to the trpezoid rule i Moreover the trpezoidl rule tends to ecome f C etremely ccurte when periodic functions re And if then we will hve the integrted over their periods which cn e error evlution : nlyzed in vrious wys n i The qudrture trpezoidl formul is: ( n) f ( d ) SD ( f) f( d ) Si( f) i i S f P f d m f ''( ) 4n [ ] Where From Newton s formul of representing of polynomil interpoltion we hve the equlity P f f f f f f
Then f f S f f d f f f f f f C If from error evlution formul t the interpoltion we hve: f P f m f '' [ ] Thus f d S f f P f d f P f d m f ''( ) d [ ] m f ''( ) m f ''( ) [ ] 6 [ ] The summed qudrture trpezoidl formul is: n n ST f ii f i f i i n f i f i n i n f i f i h i For which if the error evlution: f C n i we will hve n f ds f f ds f T m f '( ) n [ ] i i i where Si ( f) is the trpezoidl qudrture formul of the f function pplied on the i i intervl i n RESULTS AND DISCUSSIONS d ) Approimte the integrl It is very simple integrl whose primitive is clculle: 4 d 5 4 4 We will lso pproimte it with oth methods The clcultions re done with QUATTRO PRO softwre If we denote f ( ) then ' f ( ) nd f '' ( ) 6 [] Approimtion ) By rectngles method ' M m f ( ) [] We denote nd let e the error of pproimtion In order to tht M 4n it suffices tht ( ) M n 4 In our cse for we must tke n 76 = = h= 57 n= 76 Xi (Xi+Xi+)/ f((xi+xi+)/) 6578947 847568E-7 57 6578947 847568E-7 98684 9945 98977865
We otined (76) SD 49978 Approimtion ) By trpezoids method '' M m f ( ) 6 [] We denote nd let e the error of pproimtion In order tht M n it suffices tht ( ) M n In our cse for we must tke n 8 = = h= 5 n= 8 Xi f(xi) [f(xi)+f(xi+)]/ 976565 5 955 878965 5 565 4796875 75 57475 88867875 5 5 84575 65 44465 785 75 4875 545898475 875 6699875 8496975 We otin (8) ST 5965 Remrk The trpezoids method converges much fster the the rectngles method so we will do the rest of our pplictions only with the trpezoids method ln( ) d ) Approimte the integrl In this cse the primitive is inclculle (trnscendentl) so we cn t compute the integrl y elementry methods We pproimte it using the trpezoids method ln( ) f :[] f( ) Let We hve ln( ) f ''( ) ( ) ( ) In order to estimte the mimum vlue of f ''( ) we trce the grph using the CHART commnd in QUATTRO 6 4 8 6 4 f"() 45 9 99 84 4 7 5 6 6 5 7 45 8 9 9 9 8 4 6 8 Using the OPTIMIZER commnd in QUATTRO we find 5 Thus for we must tke n We otin ln( ) d ST () 6678 sin ) Approimte the integrl d 4
The primitive is trnscendentl Let sin f : f( ) We hve sin cos sin f ''( ) 4 We find M m f ''( ) [] 5 Thus for we must tke n 5 We otin The proility ho hve vlues in [ ] is P N( ) e d sin d ST (5) 859 4) Approimte the integrl The primitive is trnscendentl Let f : f() e We hve f ''( ) 4 e e d We find M m f ''( ) [] 5 Thus for we must tke n We otin () e d S T 7468 Remrk(see http://enwikipediorg/wiki/norml_distriutio n) f( ) e In sttistics is the density function of the norml distriution N( ) (with the men nd the stndrd devition ) Mny rndom vriles or fenomenons hve norml distriution whose grph is lso known s the ell of Guss: the mrks from test people s hights people s IQ etc Figure 9 The Gussin ell An integrl which cn e pproimted in the sme mnner s efore 5) Approimte the integrl sin d (n elliptic integrl) The primitive is trnscendentl Let f : f( ) ksin k We hve 4k cos (k 4 k)cos k f "( ) k kcos 4 We could hve used QUATTRO for computing the mimum ut using modul s properties we found the generl inequlity f "( ) k k So for k= M m f ''( ) [ ] 5 For we should hve tken n so for the ske of simplicity we took for which it suffices to consider n 9 We otin sin d ST (9) 968844 Remrk http://enwikipediorg/wiki/ellipse) (see 5
Using for emple line integrls the circumference of n ellipse with the semi-es is ds sin d E CONCLUSIONS We succesfully mnged to clculte (pproimtively) some importnt integrls minly with trnscendentl primitives Rectngles formul is little esier to pply esspecily if we tke the the etremities insted of the middle ut trpezoids formul is fster Although there eist even fster methods Sometimes in order to get good pproimtion we must tke ig vlule for n which mkes the prolem difficult in QUATTRO or EXCEL In this cses progrmming lnguge or routine would e etter REFERENCES Figure An ellipse Unfortuntely this integrl clled elliptic is trnscendentl so we cn only pproimte it Tking k we hve to evlute ksin d In fct we see tht our lst clcultion represents n proimtion for the circumference of the ellipse with nd Our result is very close to Rmnujn s pproimtion: C ( ) ( ) ( ) 96884 Grigore G 99 Lecţii de nliză numerică Ed Universităţii din Bucureşti ; Muntenu IP Stnică D 6 Anliză numerică Eerciţii şi teme de lortor Ed Universităţii din Bucureşti ; Roşc I Anliză numerică Ed Universităţii din Bucureşti; http://enwikipediorg/wiki/norml_distriution http://wwwmorg/eternl_rchive/jom/volume7/ak tumen/rectnglehtml http://wwwmthcsemoryedu/~cheung/courses/7/syll us/7/rectngle-methodhtml http://enwikipediorg/wiki/ellipse 6