ch1873x_p6.qx 4/12/5 11:5 AM Pge 568 PART SIX
NUMERICAL DIFFERENTIATION AND INTEGRATION PT6.1 MOTIVATION Clculus is the mthemtics of chnge. Becuse engineers must continuously el with systems n processes tht chnge, clculus is n essentil tool of our profession. Stning t the hert of clculus re the relte mthemticl concepts of ifferentition n integrtion. Accoring to the ictionry efinition, to ifferentite mens to mrk off by ifferences; istinguish;... toperceive the ifference in or between. Mthemticlly, the erivtive, which serves s the funmentl vehicle for ifferentition, represents the rte of chnge of epenent vrible with respect to n inepenent vrible. As epicte in Fig. PT6.1, the mthemticl efinition of the erivtive begins with ifference pproximtion: y x = f(x i + x) f(x i ) (PT6.1) x where y n f(x) re lterntive representtives for the epenent vrible n x is the inepenent vrible. If x is llowe to pproch zero, s occurs in moving from Fig. PT6.1 to c, the ifference becomes erivtive y x = lim x f(x i + x) f(x i ) x FIGURE PT6.1 The grphicl efinition of erivtive: s x pproches zero in going from () to (c), the ifference pproximtion becomes erivtive. f (x i + x) y y y y f (x i + x) f (x i ) f (x i ) y f'(x i ) x i x i + x x x i x i + x x x i x x () x (b) (c) 569
57 NUMERICAL DIFFERENTIATION AND INTEGRATION f (x) b x FIGURE PT6.2 Grphicl representtion of the integrl of f(x) between the limits x = to b. The integrl is equivlent to the re uner the curve. where y/x [which cn lso be esignte s y or f (x i )] is the first erivtive of y with respect to x evlute t x i. As seen in the visul epiction of Fig. PT6.1c, the erivtive is the slope of the tngent to the curve t x i. The inverse process to ifferentition in clculus is integrtion. Accoring to the ictionry efinition, to integrte mens to bring together, s prts, into whole; to unite; to inicte the totl mount.... Mthemticlly, integrtion is represente by I = b f(x) x (PT6.2) which stns for the integrl of the function f(x) with respect to the inepenent vrible x, evlute between the limits x = to x = b. The function f(x) in Eq. (PT6.2) is referre to s the integrn. As suggeste by the ictionry efinition, the mening of Eq. (PT6.2) is the totl vlue, or summtion, of f(x) x over the rnge x = to b. In fct, the symbol is ctully stylize cpitl S tht is intene to signify the close connection between integrtion n summtion. Figure PT6.2 represents grphicl mnifesttion of the concept. For functions lying bove the x xis, the integrl expresse by Eq. (PT6.2) correspons to the re uner the curve of f(x) between x = n b. 1 As outline bove, the mrking off or iscrimintion of ifferentition n the bringing together of integrtion re closely linke processes tht re, in fct, inversely 1 It shoul be note tht the process represente by Eq. (PT6.2) n Fig. PT6.2 is clle efinite integrtion. There is nother type clle inefinite integrtion in which the limits n b re unspecifie. As will be iscusse in Prt Seven, inefinite integrtion els with etermining function whose erivtive is given.
PT6.1 MOTIVATION 571 y v 4 4 2 2 4 8 12 t 4 8 12 t v y 4 4 2 2 FIGURE PT6.3 The contrst between () ifferentition n (b) integrtion. 4 () 8 12 t 4 (b) 8 12 t relte (Fig. PT6.3). For exmple, if we re given function y(t) tht specifies n object s position s function of time, ifferentition provies mens to etermine its velocity, s in (Fig. PT6.3). v(t) = t y(t) Conversely, if we re provie with velocity s function of time, integrtion cn be use to etermine its position (Fig. PT6.3b), y(t) = t v(t) t Thus, we cn mke the generl clim tht the evlution of the integrl I = b f(x) x is equivlent to solving the ifferentil eqution y x = f(x) for y(b) given the initil conition y() =. Becuse of this close reltionship, we hve opte to evote this prt of the book to both processes. Among other things, this will provie the opportunity to highlight their similrities n ifferences from numericl perspective. In ition, our iscussion will hve relevnce to the next prts of the book where we will cover ifferentil equtions.
572 NUMERICAL DIFFERENTIATION AND INTEGRATION PT6.1.1 Noncomputer Methos for Differentition n Integrtion The function to be ifferentite or integrte will typiclly be in one of the following three forms: 1. A simple continuous function such s polynomil, n exponentil, or trigonometric function. 2. A complicte continuous function tht is ifficult or impossible to ifferentite or integrte irectly. 3. A tbulte function where vlues of x n f(x) re given t number of iscrete points, s is often the cse with experimentl or fiel t. In the first cse, the erivtive or integrl of simple function my be evlute nlyticlly using clculus. For the secon cse, nlyticl solutions re often imprcticl, n sometimes impossible, to obtin. In these instnces, s well s in the thir cse of iscrete t, pproximte methos must be employe. A noncomputer metho for etermining erivtives from t is clle equl-re grphicl ifferentition. In this metho, the (x, y) t re tbulte n, for ech intervl, simple ivie ifference y/ x is employe to estimte the slope. Then these vlues re plotte s steppe curve versus x (Fig. PT6.4). Next, smooth curve is rwn tht ttempts to pproximte the re uner the steppe curve. Tht is, it is rwn so tht visully, the positive n negtive res re blnce. The rtes t given vlues of x cn then be re from the curve. In the sme spirit, visully oriente pproches were employe to integrte tbulte t n complicte functions in the precomputer er. A simple intuitive pproch is to plot the function on gri (Fig. PT6.5) n count the number of boxes tht pproximte the re. This number multiplie by the re of ech box provies rough estimte of the totl FIGURE PT6.4 Equl-re ifferentition. () Centere finite ivie ifferences re use to estimte the erivtive for ech intervl between the t points. (b) The erivtive estimtes re plotte s br grph. A smooth curve is superimpose on this plot to pproximte the re uner the br grph. This is ccomplishe by rwing the curve so tht equl positive n negtive res re blnce. (c) Vlues of y/x cn then be re off the smooth curve. x 3 6 9 15 18 y 2 35 47 65 72 () y/ x 66.7 5 4 3 23.3 y x 5 3 6 9 12 15 18 (b) x x 3 6 9 15 18 y/x 76.5 57.5 45. 36.25 25. 21.5 (c)
PT6.1 MOTIVATION 573 f (x) FIGURE PT6.5 The use of gri to pproximte n integrl. b x f (x) FIGURE PT6.6 The use of rectngles, or strips, to pproximte the integrl. b x re uner the curve. This estimte cn be refine, t the expense of itionl effort, by using finer gri. Another commonsense pproch is to ivie the re into verticl segments, or strips, with height equl to the function vlue t the mipoint of ech strip (Fig. PT6.6). The re of the rectngles cn then be clculte n summe to estimte the totl re. In this
574 NUMERICAL DIFFERENTIATION AND INTEGRATION pproch, it is ssume tht the vlue t the mipoint provies vli pproximtion of the verge height of the function for ech strip. As with the gri metho, refine estimtes re possible by using more (n thinner) strips to pproximte the integrl. Although such simple pproches hve utility for quick estimtes, lterntive numericl techniques re vilble for the sme purpose. Not surprisingly, the simplest of these methos is similr in spirit to the noncomputer techniques. For ifferentition, the most funmentl numericl techniques use finite ivie ifferences to estimte erivtives. For t with error, n lterntive pproch is to fit smooth curve to the t with technique such s lest-squres regression n then ifferentite this curve to obtin erivtive estimtes. In similr spirit, numericl integrtion or qurture methos re vilble to obtin integrls. These methos, which re ctully esier to implement thn the gri pproch, re similr in spirit to the strip metho. Tht is, function heights re multiplie by strip withs n summe to estimte the integrl. However, through clever choices of weighting fctors, the resulting estimte cn be me more ccurte thn tht from the simple strip metho. As in the simple strip metho, numericl integrtion n ifferentition techniques utilize t t iscrete points. Becuse tbulte informtion is lrey in such form, it is nturlly comptible with mny of the numericl pproches. Although continuous functions re not originlly in iscrete form, it is usully simple proposition to use the given eqution to generte tble of vlues. As epicte in Fig. PT6.7, this tble cn then be evlute with numericl metho. PT6.1.2 Numericl Differentition n Integrtion in Engineering The ifferentition n integrtion of function hs so mny engineering pplictions tht you were require to tke ifferentil n integrl clculus in your first yer t college. Mny specific exmples of such pplictions coul be given in ll fiels of engineering. Differentition is commonplce in engineering becuse so much of our work involves chrcterizing the chnges of vribles in both time n spce. In fct, mny of the lws n other generliztions tht figure so prominently in our work re bse on the preictble wys in which chnge mnifests itself in the physicl worl. A prime exmple is Newton s secon lw, which is not couche in terms of the position of n object but rther in its chnge of position with respect to time. Asie from such temporl exmples, numerous lws governing the sptil behvior of vribles re expresse in terms of erivtives. Among the most common of these re those lws involving potentils or grients. For exmple, Fourier s lw of het conuction quntifies the observtion tht het flows from regions of high to low temperture. For the one-imensionl cse, this cn be expresse mthemticlly s Het flux = k T x Thus, the erivtive provies mesure of the intensity of the temperture chnge, or grient, tht rives the trnsfer of het. Similr lws provie workble moels in mny other res of engineering, incluing the moeling of flui ynmics, mss trnsfer, chemicl rection kinetics, n electromgnetic flux. The bility to ccurtely estimte erivtives is n importnt fcet of our cpbility to work effectively in these res.
PT6.1 MOTIVATION 575 2 () 2+cos (1 + x 3/2 ) 1+.5 sin x e.5x x (b) x.25.75 1.25 1.75 f (x) 2.599 2.414 1.945 1.993 f (x) Discrete points Continuous function FIGURE PT6.7 Appliction of numericl integrtion metho: () A complicte, continuous function. (b) Tble of iscrete vlues of f(x) generte from the function. (c) Use of numericl metho (the strip metho here) to estimte the integrl on the bsis of the iscrete points. For tbulte function, the t is lrey in tbulr form (b); therefore, step () is unnecessry. (c) 2 1 1 2 x Just s ccurte estimtes of erivtives re importnt in engineering, the clcultion of integrls is eqully vluble. A number of exmples relte irectly to the ie of the integrl s the re uner curve. Figure PT6.8 epicts few cses where integrtion is use for this purpose. Other common pplictions relte to the nlogy between integrtion n summtion. For exmple, common ppliction is to etermine the men of continuous functions. In Prt Five, you were introuce to the concept of the men of n iscrete t points [recll Eq. (PT5.1)]: n i=1 y i Men = (PT6.3) n where y i re iniviul mesurements. The etermintion of the men of iscrete points is epicte in Fig. PT6.9.
576 NUMERICAL DIFFERENTIATION AND INTEGRATION () (b) (c) FIGURE PT6.8 Exmples of how integrtion is use to evlute res in engineering pplictions. () A surveyor might nee to know the re of fiel boune by menering strem n two ros. (b) A wter-resource engineer might nee to know the cross-sectionl re of river. (c) A structurl engineer might nee to etermine the net force ue to nonuniform win blowing ginst the sie of skyscrper. FIGURE PT6.9 An illustrtion of the men for () iscrete n (b) continuous t. y Men 1 2 3 4 5 6 () i y = f (x) Men (b) b x
PT6.1 MOTIVATION 577 In contrst, suppose tht y is continuous function of n inepenent vrible x, se- picte in Fig. PT6.9b. For this cse, there re n infinite number of vlues between n b. Just s Eq. (PT6.3) cn be pplie to etermine the men of the iscrete reings, you might lso be intereste in computing the men or verge of the continuous function y = f(x) for the intervl from to b. Integrtion is use for this purpose, s specifie by the formul Men = b f(x) x b (PT6.4) This formul hs hunres of engineering pplictions. For exmple, it is use to clculte the center of grvity of irregulr objects in mechnicl n civil engineering n to etermine the root-men-squre current in electricl engineering. Integrls re lso employe by engineers to evlute the totl mount or quntity of given physicl vrible. The integrl my be evlute over line, n re, or volume. For exmple, the totl mss of chemicl contine in rector is given s the prouct of the concentrtion of chemicl n the rector volume, or Mss = concentrtion volume where concentrtion hs units of mss per volume. However, suppose tht concentrtion vries from loction to loction within the rector. In this cse, it is necessry to sum the proucts of locl concentrtions c i n corresponing elementl volumes V i : Mss = n c i V i i=1 where n is the number of iscrete volumes. For the continuous cse, where c(x, y, z) is known function n x, y, n z re inepenent vribles esignting position in Crtesin coorintes, integrtion cn be use for the sme purpose: Mss = c(x, y, z) x yz or Mss = V c(v ) V which is referre to s volume integrl. Notice the strong nlogy between summtion n integrtion. Similr exmples coul be given in other fiels of engineering. For exmple, the totl rte of energy trnsfer cross plne where the flux (in clories per squre centimeter per secon) is function of position is given by Het trnsfer = flux A A which is referre to s n rel integrl, where A = re.
578 NUMERICAL DIFFERENTIATION AND INTEGRATION Similrly, for the one-imensionl cse, the totl mss of vrible-ensity ro with constnt cross-sectionl re is given by m = A L ρ(x) x where m = totl weight (kg), L = length of the ro (m), ρ(x) = known ensity (kg/m 3 ) s function of length x (m), n A = cross-sectionl re of the ro (m 2 ). Finlly, integrls re use to evlute ifferentil or rte equtions. Suppose the velocity of prticle is known continuous function of time v(t), y = v(t) t The totl istnce y trvele by this prticle over time t is given by (Fig. PT6.3b) y = t v(t) t (PT6.5) These re just few of the pplictions of ifferentition n integrtion tht you might fce regulrly in the pursuit of your profession. When the functions to be nlyze re simple, you will normlly choose to evlute them nlyticlly. For exmple, in the flling prchutist problem, we etermine the solution for velocity s function of time [Eq. (1.1)]. This reltionship coul be substitute into Eq. (PT6.5), which coul then be integrte esily to etermine how fr the prchutist fell over time perio t. For this cse, the integrl is simple to evlute. However, it is ifficult or impossible when the function is complicte, s is typiclly the cse in more relistic exmples. In ition, the unerlying function is often unknown n efine only by mesurement t iscrete points. For both these cses, you must hve the bility to obtin pproximte vlues for erivtives n integrls using numericl techniques. Severl such techniques will be iscusse in this prt of the book. PT6.2 MATHEMATICAL BACKGROUND In high school or uring your first yer of college, you were introuce to ifferentil n integrl clculus. There you lerne techniques to obtin nlyticl or exct erivtives n integrls. When we ifferentite function nlyticlly, we generte secon function tht cn be use to compute the erivtive for ifferent vlues of the inepenent vrible. Generl rules re vilble for this purpose. For exmple, in the cse of the monomil y = x n the following simple rule pplies (n ): y x = nxn 1 which is the expression of the more generl rule for y = u n
PT6.2 MATHEMATICAL BACKGROUND 579 where u = function of x. For this eqution, the erivtive is compute vi y x = nun 1 u x Two other useful formuls pply to the proucts n quotients of functions. For exmple, if the prouct of two functions of x(u n v) is represente s y = uv, then the erivtive cn be compute s y x = u v x + v u x For the ivision, y = u/v, the erivtive cn be compute s u v y x = x u v x v 2 Other useful formuls re summrize in Tble PT6.1. Similr formuls re vilble for efinite integrtion, which els with etermining n integrl between specifie limits, s in I = b f(x) x Accoring to the funmentl theorem of integrl clculus, Eq. (PT6.6) is evlute s b f(x) x = F(x) b (PT6.6) where F(x) = the integrl of f(x) tht is, ny function such tht F (x) = f(x). The nomenclture on the right-hn sie stns for F(x) b = F(b) F() (PT6.7) TABLE PT6.1 Some commonly use erivtives. sin x = cos x x cos x = sin x x tn x = sec 2 x x ln x = 1 x x e x = e x x cot x = csc 2 x x sec x = sec x tn x x csc x = csc x cot x x 1 log x = x x I n x = x ln x
58 NUMERICAL DIFFERENTIATION AND INTEGRATION An exmple of efinite integrl is I =.8 (.2 + 25x 2x 2 + 675x 3 9x 4 + 4x 5 ) x (PT6.8) For this cse, the function is simple polynomil tht cn be integrte nlyticlly by evluting ech term ccoring to the rule b x n x = xn+1 b n + 1 (PT6.9) where n cnnot equl 1. Applying this rule to ech term in Eq. (PT6.8) yiels I =.2x + 12.5x 2 2 3 x3 + 168.75x 4 18x 5 + 4.8 6 x6 (PT6.1) which cn be evlute ccoring to Eq. (PT6.7) s I = 1.645333. This vlue is equl to the re uner the originl polynomil [Eq. (PT6.8)] between x = n.8. The foregoing integrtion epens on knowlege of the rule expresse by Eq. (PT6.9). Other functions follow ifferent rules. These rules re ll merely instnces of ntiifferentition, tht is, fining F(x) so tht F (x) = f(x). Consequently, nlyticl integrtion epens on prior knowlege of the nswer. Such knowlege is cquire by trining n TABLE PT6.2 Some simple integrls tht re use in Prt Six. The n b in this tble re constnts n shoul not be confuse with the limits of integrtion iscusse in the text. u v= uv v u n+1 u n u u = + C n 1 n 1 bx bx x = + C >, 1 b In x = ln x +C x x sin (x + b) x = 1 cos (x + b) + C cos (x + b) x = 1 sin (x + b) + C ln x x = x ln x x + C e x x = e x x + C xe x x = e 2 (x 1) + C 1 xbx 2 = b tn 1 b x + C
PT6.3 ORIENTATION 581 experience. Mny of the rules re summrize in hnbooks n in tbles of integrls. We list some commonly encountere integrls in Tble PT6.2. However, mny functions of prcticl importnce re too complicte to be contine in such tbles. One reson why the techniques in the present prt of the book re so vluble is tht they provie mens to evlute reltionships such s Eq. (PT6.8) without knowlege of the rules. PT6.3 ORIENTATION Before proceeing to the numericl methos for integrtion, some further orienttion might be helpful. The following is intene s n overview of the mteril iscusse in Prt Six. In ition, we hve formulte some objectives to help focus your efforts when stuying the mteril. PT6.3.1 Scope n Preview Figure PT6.1 provies n overview of Prt Six. Chpter 21 is evote to the most common pproches for numericl integrtion the Newton-Cotes formuls. These reltionships re bse on replcing complicte function or tbulte t with simple polynomil tht is esy to integrte. Three of the most wiely use Newton-Cotes formuls re iscusse in etil: the trpezoil rule, Simpson s 1/3 rule, n Simpson s 3/8 rule. All these formuls re esigne for cses where the t to be integrte is evenly spce. In ition, we lso inclue iscussion of numericl integrtion of uneqully spce t. This is very importnt topic becuse mny rel-worl pplictions el with t tht is in this form. All the bove mteril reltes to close integrtion, where the function vlues t the ens of the limits of integrtion re known. At the en of Chp. 21, we present open integrtion formuls, where the integrtion limits exten beyon the rnge of the known t. Although they re not commonly use for efinite integrtion, open integrtion formuls re presente here becuse they re utilize extensively in the solution of orinry ifferentil equtions in Prt Seven. The formultions covere in Chp. 21 cn be employe to nlyze both tbulte t n equtions. Chpter 22 els with two techniques tht re expressly esigne to integrte equtions n functions: Romberg integrtion n Guss qurture. Computer lgorithms re provie for both of these methos. In ition, methos for evluting improper integrls re iscusse. In Chp. 23, we present itionl informtion on numericl ifferentition to supplement the introuctory mteril from Chp. 4. Topics inclue high-ccurcy finite-ifference formuls, Richrson s extrpoltion, n the ifferentition of uneqully spce t. The effect of errors on both numericl ifferentition n integrtion is iscusse. Finlly, the chpter is conclue with escription of the ppliction of severl softwre pckges n librries for integrtion n ifferentition. Chpter 24 emonstrtes how the methos cn be pplie for problem solving. As with other prts of the book, pplictions re rwn from ll fiels of engineering. A review section, or epilogue, is inclue t the en of Prt Six. This review inclues iscussion of tre-offs tht re relevnt to implementtion in engineering prctice. In ition, importnt formuls re summrize. Finlly, we present short review of vnce
582 NUMERICAL DIFFERENTIATION AND INTEGRATION PT 6.1 Motivtion PT 6.2 Mthemticl bckgroun PT 6.3 Orienttion PT 6.5 Importnt formuls PT 6.4 Tre-offs PT 6.6 Avnce methos EPILOGUE PART 6 Numericl Integrtion n Differentition 21.1 Trpezoil rule CHAPTER 21 Newton-Cotes Integrtion Formuls 21.2 Simpson's rules 21.3 Unequl segments 21.4 Open integrtion 21.5 Multiple integrls 24.4 Mechnicl engineering 22.1 Newton-Cotes for equtions 24.3 Electricl engineering 24.2 Civil engineering CHAPTER 24 Engineering Cse Stuies 24.1 Chemicl engineering 23.5 Librries n pckges CHAPTER 23 Numericl Differentition CHAPTER 22 Integrtion of Equtions 23.1 High-ccurcy formuls 22.4 Improper integrls 22.2 Rhomberg integrtion 22.3 Guss qurture 23.4 Uncertin t 23.3 Unequl-spce t 23.2 Richrson extrpoltion FIGURE PT6.1 Schemtic of the orgniztion of mteril in Prt Six: Numericl Integrtion n Differentition. methos n lterntive references tht will fcilitte your further stuies of numericl ifferentition n integrtion. PT6.3.2 Gols n Objectives Stuy Objectives. After completing Prt Six, you shoul be ble to solve mny numericl integrtion n ifferentition problems n pprecite their ppliction for engineering
PT6.3 ORIENTATION 583 TABLE PT6.3 Specific stuy objectives for Prt Six. 1. Unerstn the erivtion of the Newton-Cotes formuls; know how to erive the trpezoil rule n how to set up the erivtion of both of Simpson s rules; recognize tht the trpezoil n Simpson s 1/3 n 3/8 rules represent the res uner first-, secon-, n thir-orer polynomils, respectively. 2. Know the formuls n error equtions for () the trpezoil rule, (b) the multiple-ppliction trpezoil rule, (c) Simpson s 1/3 rule, () Simpson s 3/8 rule, n (e) the multiple-ppliction Simpson s rule. Be ble to choose the best mong these formuls for ny prticulr problem context. 3. Recognize tht Simpson s 1/3 rule is fourth-orer ccurte even though it is bse on only three points; relize tht ll the even-segment o-point Newton-Cotes formuls hve similr enhnce ccurcy. 4. Know how to evlute the integrl n erivtive of uneqully spce t. 5. Recognize the ifference between open n close integrtion formuls. 6. Unerstn the theoreticl bsis of Richrson extrpoltion n how it is pplie in the Romberg integrtion lgorithm n for numericl ifferentition. 7. Unerstn the funmentl ifference between Newton-Cotes n Guss qurture formuls. 8. Recognize why both Romberg integrtion n Guss qurture hve utility when integrting equtions (s oppose to tbulr or iscrete t). 9. Know how open integrtion formuls re employe to evlute improper integrls. 1. Unerstn the ppliction of high-ccurcy numericl-ifferentition formuls. 11. Know how to ifferentite uneqully spce t. 12. Recognize the iffering effects of t error on the processes of numericl integrtion n ifferentition. problem solving. You shoul strive to mster severl techniques n ssess their relibility. You shoul unerstn the tre-offs involve in selecting the best metho (or methos) for ny prticulr problem. In ition to these generl objectives, the specific concepts liste in Tble PT6.3 shoul be ssimilte n mstere. Computer Objectives. You hve been provie with softwre n simple computer lgorithms to implement the techniques iscusse in Prt Six. All hve utility s lerning tools. Algorithms re provie for most of the other methos in Prt Six. This informtion will llow you to expn your softwre librry to inclue techniques beyon the trpezoil rule. For exmple, you my fin it useful from professionl viewpoint to hve softwre to implement numericl integrtion n ifferentition of uneqully spce t. You my lso wnt to evelop your own softwre for Simpson s rules, Romberg integrtion, n Guss qurture, which re usully more efficient n ccurte thn the trpezoil rule. Finlly, one of your most importnt gols shoul be to mster severl of the generlpurpose softwre pckges tht re wiely vilble. In prticulr, you shoul become ept t using these tools to implement numericl methos for engineering problem solving.