Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego wth curved oudry. We hve see tht ths prolem hd ee solved tquty for prtculr cses, such s the crcle. But to cheve our gol we hve to move towrds more geerl stuto. We shll eg to do so y cosderg regos tht re lmost rectgles, except for oe curved sde. Here s more forml descrpto of wht we shll lyze. Wht you c ler here: The key method for pproxmtg the re of curved rego, whch wll led to the method for computg t exctly. Such rego wll look lke ths: y f x Defto By the re uder curve, we me the re of the rego ouded: y the x-xs elow y the vertcl le x vertcl le x to the rght, wth to the left d the y the postve d cotuous fucto y f x ove. To pproxmte ts re, we use Archmedes de, ut sted of dvdg the rego to sectors, s he dd for the crcle, we dvde t to vertcl slces. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 1
y f x Notce tht, ulke wht hppeed to Archmedes wth the crcle, here t s ot cler wht heght we should ssg to ech rectgle so s to hve resole pproxmto. We hve severl choces, some of whch wll e detfed specfclly the ext secto. For ow, we shll focus o orgzg the otto, so s to e le to work wth t effcetly. Notto for the pproxmto of the re uder curve We the pproxmte the re of ech slce y usg rectgles. y f x We lel the pproxmtg rectgles from the left to the rght s R1, R2,..., R, ech hvg wdth w d heght h, 1, 2,...,. For coveece we shll dvde the tervl, to tervls of equl legth, so tht w x for ll rectgles. The heght of the -th pproxmtg rectgle s f x for some vlue x the -th tervl. The re of the -th pproxmtg rectgle s A w f x, or, wth therefore represeted y equl wdths, A f x x. The sum of the res of the pproxmtg rectgles s gve y A f x w, or, 1. 1 wth equl wdths, A f x x Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 2
Ths otto s demostrted these pctures. h f x f x 2 f x R 2 h 2 R h f x 1 h 1 R R 1 w w w w 1 2 w x x1 x2 x x The formul tht we shll use to pproxmte the whole re s key pece of formto tht wll e used repetedly. It s very mportt d t hs ts ow me. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 3
A qutty of the form: Defto 1 f x x y f x s clled Rem sum of the fucto o the tervl, d t provdes pproxmto to the re uder tht fucto o the sme tervl. I prtculr, whe usg tervls of equl wdth, the formul: f x x, x 1 provdes Rem sum for the fucto. Wth these choces, the Rem sum we ot s: 16 1.2 1.2 16 2 2 16 3.5 3.5 16 4.6 4.6 2 2 2 2 Its vlue s 32.95. Boy, dd tht tke lot of steps, d for esy cse! But we ll soo develop etter wy. Notce tht Rem sum ccomplshes the gol of pproxmtg the re uder the curve d t does so wy tht llows for etter d etter pproxmtos smply y usg greter umer of slces, ll wth smller wdths. But we hve ot solved the re prolem, sce we stll hve some ssues to resolve: 1) Sce we eed to mke choce of strps d represettve pots, we ed up wth dfferet pproxmtos for ech such choce, so we eed to ssess the propertes of such possle optos. 2) Wht we relly wt s the exct vlue of such re, ot just pproxmto. 3) The otto we re uldg s very cumersome d eeds to e refed. Strtg from the ext secto, we shll ddress ll these ssues d come up, thks to Newto, Lez, Rem d my other mthemtcs, wth very workle soluto to the geerl prolem. I shll coclude ths secto y metog some mes ssocted wth specl choces of the vlues ech tervl. f x 1 6 x x, 1, 5 Exmple: 2 We c costruct Rem sum for ths rego y usg 4 d tervls of 1, 2, 2, 3, 3, 4, 4, 5. equl wdths: Ths gves: 4 1 4 1 f x x 1 6x x 2 51 4 Wth ech tervl we pck vlue for x sy x 1.2, 2, 3.5, 4.6. Notce tht we c pck these vlues s we wsh! Defto If we deote y x the vlue we choose the -th tervl, the: A left Rem sum s oted y choosg x to e the left ed pot of the -th tervl. A rght Rem sum s oted y choosg x to e the rght ed pot of the -th tervl. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 4 A mdpot Rem sum s oted y
choosg x to e the mdpot of the -th tervl. A upper Rem sum s oted y choosg x to e the pot of the -th tervl for whch f x s lrgest. Although ths termology s used frequetly whe workg wth Rem sums, we shll ot use t much, sce we wll soo leve Rem sums ehd us. A lower Rem sum s oted y choosg x to e the pot of the -th tervl for whch f x s smllest. Summry The re uder curve c e well pproxmted y Rem sum cosstg of the re of th rectgles, ech pproxmtg slce of the rego questo. There re severl possle Rem sums, ech lked to prtculr choce of how the slces re costructed. Commo errors to vod Py tteto to the otto we re usg, sce t wll ecome the most mportt ssue wht comes lter. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 5
Lerg questos for Secto I 4-2 Revew questos: 1. Descre wht Rem sum for fucto over tervl s. 2. Expl the role of the Rem sum the process of solvg the re prolem. Memory questos: 1. Wht sc geometrc shpe s used whe settg up the Rem sum formul? 2. Whch formul represets Rem sum? Computto questos: 1. Costruct Rem sum tht pproxmtes the re uder the curve y s x o 0, y usg 6 slces. 2. Costruct Rem sum tht pproxmtes the re uder the curve y l x o 1, 6 y usg 5 slces. 3. Costruct the geerl summto formul tht specfclly descres the pproxmto to the re ouded y the x- xs d the fucto y x l x etwee x=1 d x=e. Also, expl wht ech prt of such formul represets geometrclly. 2 x 4. Estmte the re of the rego ouded y the fucto y e e the frst qudrt y usg 4 scred rectgles. Clerly preset the specfc formul descrg the pproxmto: pure clcultor work wll ot gve you my mrks. 5. Compute Rem sum wth =6 d mdpot estmtes to pproxmte the 2 re of the rego etwee the x-xs d the fucto y cosh x etwee x=1 d x=2.5. 6. Wrte expresso tht provdes pproxmto to the re of the rego 2 y e x ouded y, the x-xs, the y-xs d the le x=5. Also, expl the prctcl (geometrc) meg of ech elemet of such expresso. I questos 7-8, detfy rego whose re c e pproxmted y the gve summto. Clerly expl the rtole you use to detfy the key fetures of ths rego. 9 7. cosh 2 1 3 8. 8 4 3e 5 1 Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 6
Theory questos: 1. Idetfy three fctors tht cotrute to mkg Rem sum pproxmto of the rel re ppled prolem. 2. How c oe mprove the pproxmto gve y Rem sum? 3. I the Rem sum, wht s the geometrcl meg of the expresso sde the sgm otto? 4. Is t ecessry to hve strps of equl wdth to costruct Rem sum? Applcto questos: 1. The ler desty of metl r s represeted y the fucto whose grph s gve here. Expl how we c use pproxmtg rectgles to estmte the re uder ths curve, use your method to ot such estmte, d determe wht the physcl meg of such re s relto to the r. 2. A pece of property s ouded y three rods, represeted y the x d y xes d the le 3 Use proper sgm otto d 6 rectgulr strps to estmte the re of such property. x, d rver whose course s represeted y the curve elow. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 7
Templted questos: 1. Wrte the correct sgm otto for y Rem sum you ecouter. 2. I ech cse where you used Rem sum to estmte re, try to determe f you oted overestmte or uderestmte. However, keep md tht ths cot lwys e doe. Wht questos do you hve for your structor? Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2: Rem sums Pge 8