In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is

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1 Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I evlutg xdx the Fudmetl Theorem o Clculus we must hve tdervtve or the tegrd x I some cses t s dcult, or eve mpossle, to d such tdervtve I such cses, pproxmte (or umercl) tegrto techque c e employed These pproxmto methods re lso used my pplcto prolems where there s o explct ormul or the ucto o terest II Let- d Rght-hd Rem Sums L d R I Clculus I you lered pproxmto method usg Rem sum Recll tht the Rem sum s ormed y dvdg the tervl [, ] to sutervls o equl wdth x We let * * * * x 0, x, x,, x e the edpots o the sutervls d choose smple pots x, x, x,, x these sutervls, so tht x les the th sutervl x, The dete tegrl s the pproxmted s * xdx x x x Tkg the let- or rght-hd edpots o ech sutervl s smple pots s smple pproch ormultg Rem sum I dog so, we ot the ollowg pproxmtos: xdx L x x xdx R x x usg The let- d rght-hd sums pproxmte the re uder the grph s the sum o the res o rectgles hvg wdth x, respectvely, s show ove Notce the let-hd sums yeld d heght x or x low estmte o the re uder x whe the ucto s cresg the hgh estmte o the re whe the ucto s decresg The opposte occurs or rght-hd sums -

2 III The Mdpot Rule M Aother commoly used smple pot or Rem sum pproxmto s the mdpot o ech sutervl, whch s deoted x Here the Rem sum s ormed y dvdg the tervl [, ] to sutervls o equl wdth x d x x x s used to ot the heght o ech rectgle The pproxmto s: xdx M x x The mdpot rule pproxmtes the re uder the grph o x rom to s the sum o the res o rectgles hvg wdth x d heght x s see the gure to the rght Notce rom the gure tht the mdpot rule provdes etter pproxmto to the dete tegrl th the let- or rght-hd sums wth the sme umer o sudvsos, Exmple : For the tegrl 00 x dx, () clculte the pproxmto M Roud your swer to our decml plces () sketch the rectgles correspodg to L, R, d M Whch pproxmto s overestmte? ) L ) R ) M -

3 IV The Trpezod Rule T The prevous methods o pproxmto use the sum o the res o rectgles The res o trpezods my lso e summed to pproxmte dete tegrl I ths cse, the tervl [, ] s, oce g, dvded to sutervls o equl wdth x We let x 0, x, x,, x e the edpots o the sutervls The dete tegrl s the pproxmted s ( x x xdx T x d s depcted elow We re mlr wth the ormul or the re o trpezod: A h where h s the heght d d re the legths o the ech sde I the pproxmto ormul ove h x, x, d or ech trpezod x uder x The summto ormul ove lso revels other mportt property o the trpezod rule It s the verge o the let- d rght-hd sums L R Tht s, T The ormul used or the Trpezod Rule s T x x x x x V Smpso s Rule S x 0 Smpso s rule ders rom the other methods tht t uses prols sted o strght-le segmets to pproxmte the re uder the ucto over ech sutervl I ths cse, the tervl [, ] s g dvded to sutervls (ut ow must e eve) o equl wdth x d x 0, x, x,, x re the edpots o the sutervls The o ech cosecutve pr o sutervls the ucto s pproxmted y prol tht s orced to mtch the vlue o x t the two edpots d the mdpot o the sutervl The dete tegrl s the pproxmted: x xdx S x x x x x x 0 It s terestg to ote tht the pproxmto usg Smpso s Rule hs weghted verge reltoshp wth the Mdpot d Trpezod rules s ollows: S M T *Note: Whe usg y o these pproxmto methods o rw dt such s tht oud tle, use exct tle vlues Do ot terpolte -

4 Exmple : For the tegrl 00 x dx, () clculte the pproxmto T usg the ormul ove () sketch the trpezods used the T pproxmto Does T yeld low or hgh estmte? Exmple : () Approxmte the dete tegrl 00 x dx usg Smpso s rule wth = 6 () Show ths pproxmto stses the weghted verge equto ove usg M d T oud prevously -

5 VI Approxmtg Errors I pplyg pproxmto techques, error s volved ( lmost every cse) The exct error volved s the derece the ctul vlue o the dete tegrl d the pproxmte vlue Tht s, A s some Error pproxmte vlue oted usg sutervls, the x dx A Geerlly, cresg mproves the ccurcy o the pproxmto Ths leds to the questo o how lrge to tke There re some error oud ormuls tht llow us to determe the gretest expected error or desred O course, gve mxmum llowle error, oe could lso determe the smllest tht would yeld such error The errors or the Trpezod d Mdpot Rules, respectvely, re: E T K d E M K " where x K or x Here, K s the gretest extrem ( solute vlue) o the secod dervtve o the tegrd o the tervl [, ] Exmple : Suppose the tegrl 0 e x dx s pproxmted usg ether the Trpezod or Mdpot rule () Expl why K < 8 mght e used or the error oud ormul s t pples to e x dx (Ht: e ) 0 () Determe the mxmum error curred pproxmtg e x dx usg M 0 (c) I the mxmum error tolerce usg T to pproxmte 0 should e used? e x dx s 000, wht s the smllest tht -

6 Mth S8 L Exercses Nme: Secto: Score: You my use your textook, lecture d l otes to help you complete ths l No other ssstce s llowed Show some work ut you my mke l clcultos usg clcultor Clculte the pproxmto M or dx Ret decml plces x () Clculte the exct vlue o dx Evlute to the proper tdervtve the use x clcultor or the vlue rouded to decml plces () Clculte the ctul error mde pproxmtg ths dete tegrl usg M (c) Clculte the mxmum expected error usg M You my use your clcultor to grph to determe the exct vlue o K x -6

7 For the tegrl x dx, 0 () Approxmte the vlue usg Trpezod Rule wth = () Determe the umer o sutervls eeded or T to esure error o greter th 00 E T K The perormce o electrc cr s studed d the ollowg veloctes, vt oted t oe secod tervls over the tme t = 0 to t = 6 secods t s, re t 0 6 v(t) Fd the est pproxmto or the dstce the cr trveled rom t = 0 to t = 6 usg ech o the ollowg methods You must determe the pproprte umer o sudvsos,, or ech () Mdpot Rule () Smpso s Rule -7