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1 SECTION Area 9 Sectio Area Use sigma otatio to write ad evaluate a sum Uderstad the cocept o area Approimate the area o a plae regio Fid the area o a plae regio usig limits Sigma Notatio I the precedig sectio, ou studied atidieretiatio I this sectio, ou will look urther ito a prolem itroduced i Sectio that o idig the area o a regio i the plae At irst glace, these two ideas ma seem urelated, ut ou will discover i Sectio that the are closel related a etremel importat theorem called the Fudametal Theorem o Calculus This sectio egis itroducig a cocise otatio or sums This otatio is called sigma otatio ecause it uses the uppercase Greek letter sigma, writte as Sigma Notatio The sum o terms a, a, a,, a is writte as a i a a a a where i is the ide o summatio, a i is the ith term o the sum, ad the upper ad lower ouds o summatio are ad NOTE The upper ad lower ouds must e costat with respect to the ide o summatio However, the lower oud does t have to e A iteger less tha or equal to the upper oud is legitimate EXAMPLE Eamples o Sigma Notatio FOR FURTHER INFORMATION For a geometric iterpretatio o summatio ormulas, see the article, Lookig at k ad k Geometricall Eric k k Heglom i Mathematics Teacher MathArticle a 6 i 6 i 6 i0 c 7 j 6 7 j d k k e i From parts (a) ad (), otice that the same sum ca e represeted i dieret was usig sigma otatio Tr It Eploratio A Techolog Although a variale ca e used as the ide o summatio i, j, ad k are ote used Notice i Eample that the ide o summatio does ot appear i the terms o the epaded sum

2 60 CHAPTER Itegratio THE SUM OF THE FIRST 00 INTEGERS Carl Friedrich Gauss s (777 8) teacher asked him to add all the itegers rom to 00 Whe Gauss retured with the correct aswer ater ol a ew momets, the teacher could ol look at him i astouded silece This is what Gauss did: This is geeralized Theorem, where 00 i t 00 0 The ollowig properties o summatio ca e derived usig the associative ad commutative properties o additio ad the distriutive propert o additio over multiplicatio (I the irst propert, k is a costat) ka i k a i a i ± i a i ± i The et theorem lists some useul ormulas or sums o powers A proo o this theorem is give i Appedi A THEOREM Summatio Formulas c c i i 6 i EXAMPLE Evaluatig a Sum i Evaluate or 0, 00, 000, ad 0,000 i , , Solutio Applig Theorem, ou ca write i i i Factor costat out o sum Write as two sums Appl Theorem Simpli Simpli Now ou ca evaluate the sum sustitutig the appropriate values o, as show i the tale at the let Tr It Eploratio A Eploratio B I the tale, ote that the sum appears to approach a limit as icreases Although the discussio o limits at iiit i Sectio applies to a variale, where ca e a real umer, ma o the same results hold true or limits ivolvig the variale, where is restricted to positive iteger values So, to id the limit o as approaches iiit, ou ca write lim

3 SECTION Area 6 Rectagle: A h Figure h Area I Euclidea geometr, the simplest tpe o plae regio is a rectagle Although people ote sa that the ormula or the area o a rectagle is A h, as show i Figure, it is actuall more proper to sa that this is the deiitio o the area o a rectagle From this deiitio, ou ca develop ormulas or the areas o ma other plae regios For eample, to determie the area o a triagle, ou ca orm a rectagle whose area is twice that o the triagle, as show i Figure 6 Oce ou kow how to id the area o a triagle, ou ca determie the area o a polgo sudividig the polgo ito triagular regios, as show i Figure 7 h Triagle: A h Figure 6 Parallelogram Heago Polgo Figure 7 ARCHIMEDES (87 BC) Archimedes used the method o ehaustio to derive ormulas or the areas o ellipses, paraolic segmets, ad sectors o a spiral He is cosidered to have ee the greatest applied mathematicia o atiquit MathBio Fidig the areas o regios other tha polgos is more diicult The aciet Greeks were ale to determie ormulas or the areas o some geeral regios (pricipall those ouded coics) the ehaustio method The clearest descriptio o this method was give Archimedes Essetiall, the method is a limitig process i which the area is squeezed etwee two polgos oe iscried i the regio ad oe circumscried aout the regio For istace, i Figure 8 the area o a circular regio is approimated a -sided iscried polgo ad a -sided circumscried polgo For each value o the area o the iscried polgo is less tha the area o the circle, ad the area o the circumscried polgo is greater tha the area o the circle Moreover, as icreases, the areas o oth polgos ecome etter ad etter approimatios o the area o the circle FOR FURTHER INFORMATION For a alterative developmet o the ormula or the area o a circle, see the article Proo Without Words: Area o a Disk is R Russell Ja Hedel i Mathematics Magazie = 6 The ehaustio method or idig the area o a circular regio Figure 8 Aimatio = MathArticle A process that is similar to that used Archimedes to determie the area o a plae regio is used i the remaiig eamples i this sectio

4 6 CHAPTER Itegratio The Area o a Plae Regio Recall rom Sectio that the origis o calculus are coected to two classic prolems: the taget lie prolem ad the area prolem Eample egis the ivestigatio o the area prolem EXAMPLE Approimatig the Area o a Plae Regio () = (a) The area o the paraolic regio is greater tha the area o the rectagles () = () The area o the paraolic regio is less tha the area o the rectagles Figure 9 Use the ive rectagles i Figure 9(a) ad () to id two approimatios o the area o the regio lig etwee the graph o ad the -ais etwee 0 ad Solutio a The right edpoits o the ive itervals are i, where i,,,, The width o each rectagle is, ad the height o each rectagle ca e otaied evaluatig at the right edpoit o each iterval 0,,,,, 6, 6, 8, 8, 0 Evaluate at the right edpoits o these itervals The sum o the areas o the ive rectagles is Height Width i i 6 68 Because each o the ive rectagles lies iside the paraolic regio, ou ca coclude that the area o the paraolic regio is greater tha 68 The let edpoits o the ive itervals are i, where i,,,, The width o each rectagle is, ad the height o each rectagle ca e otaied evaluatig at the let edpoit o each iterval Height Width i i Because the paraolic regio lies withi the uio o the ive rectagular regios, ou ca coclude that the area o the paraolic regio is less tha 808 B comiig the results i parts (a) ad (), ou ca coclude that 68 < Area o regio < 808 Tr It Eploratio A Eploratio B Video NOTE B icreasig the umer o rectagles used i Eample, ou ca otai closer ad closer approimatios o the area o the regio For istace, usig rectagles o width each, ou ca coclude that 77 < Area o regio < 79

5 SECTION Area 6 Upper ad Lower Sums The procedure used i Eample ca e geeralized as ollows Cosider a plae regio ouded aove the graph o a oegative, cotiuous uctio, as show i Figure 0 The regio is ouded elow the -ais, ad the let ad right oudaries o the regio are the vertical lies a ad To approimate the area o the regio, egi sudividig the iterval a, ito suitervals, each o width a, as show i Figure The edpoits o the itervals are as ollows a 0 a The regio uder a curve Figure 0 (m i ) a The iterval a, is divided ito suitervals o width a Figure (M i ) a 0 < a < a < < a Because is cotiuous, the Etreme Value Theorem guaratees the eistece o a miimum ad a maimum value o i each suiterval m i Miimum value o i ith suiterval M i Maimum value o i ith suiterval Net, deie a iscried rectagle lig iside the ith suregio ad a circumscried rectagle etedig outside the ith suregio The height o the ith iscried rectagle is m i ad the height o the ith circumscried rectagle is M i For each i, the area o the iscried rectagle is less tha or equal to the area o the circumscried rectagle Area o iscried rectagle The sum o the areas o the iscried rectagles is called a lower sum, ad the sum o the areas o the circumscried rectagles is called a upper sum Lower sum s Upper sum S m i M i Area o iscried rectagles Area o circumscried rectagles From Figure, ou ca see that the lower sum s is less tha or equal to the upper sum S Moreover, the actual area o the regio lies etwee these two sums s Area o regio S m i M i Area o circumscried rectagle s() = () = () S() = () a a Area o iscried rectagles Area o regio Area o circumscried is less tha area o regio rectagles is greater tha area o regio Figure a As icreases, oth the lower sum s ad the upper sum S ecome closer to the actual area o the regio View the aimatio to see this Aimatio

6 6 CHAPTER Itegratio EXAMPLE Fidig Upper ad Lower Sums or a Regio () = Iscried rectagles () = Circumscried rectagles Figure Editale Graph Fid the upper ad lower sums or the regio ouded the graph o ad the -ais etwee 0 ad Solutio To egi, partitio the iterval 0, ito suitervals, each o width a Figure shows the edpoits o the suitervals ad several iscried ad circumscried rectagles Because is icreasig o the iterval 0,, the miimum value o each suiterval occurs at the let edpoit, ad the maimum value occurs at the right edpoit Let Edpoits m i 0 i i Usig the let edpoits, the lower sum is s Usig the right edpoits, the upper sum is S 0 m i i i M i 8 i i 8 i i i 8 i Right Edpoits M i 0 i i i Lower sum Upper sum Tr It Eploratio A Eploratio B

7 SECTION Area 6 EXPLORATION For the regio give i Eample, evaluate the lower sum s 8 ad the upper sum S 8 or 0, 00, ad 000 Use our results to determie the area o the regio Eample illustrates some importat thigs aout lower ad upper sums First, otice that or a value o, the lower sum is less tha (or equal to) the upper sum s 8 < 8 S Secod, the dierece etwee these two sums lesses as icreases I act, i ou take the limits as 8, oth the upper sum ad the lower sum approach lim s lim 8 8 lim S lim 8 8 Lower sum limit Upper sum limit The et theorem shows that the equivalece o the limits (as ) o the upper ad lower sums is ot mere coicidece It is true or all uctios that are cotiuous ad oegative o the closed iterval a, The proo o this theorem is est let to a course i advaced calculus THEOREM Limits o the Lower ad Upper Sums Let e cotiuous ad oegative o the iterval a, The limits as o oth the lower ad upper sums eist ad are equal to each other That is, lim s lim m i lim M i lim S where a ad m i ad M i are the miimum ad maimum values o o the suiterval Because the same limit is attaied or oth the miimum value m i ad the maimum value M i, it ollows rom the Squeeze Theorem (Theorem 8) that the choice o i the ith suiterval does ot aect the limit This meas that ou are ree to choose a aritrar -value i the ith suiterval, as i the ollowig deiitio o the area o a regio i the plae (c i ) a c i i i The width o the ith suiterval is i i Figure Deiitio o the Area o a Regio i the Plae Let e cotiuous ad oegative o the iterval a, The area o the regio ouded the graph o, the -ais, ad the vertical lies a ad is Area lim c i, where a (see Figure ) c i i Video

8 66 CHAPTER Itegratio EXAMPLE Fidig Area the Limit Deiitio Fid the area o the regio ouded the graph, the -ais, ad the vertical lies 0 ad, as show i Figure () = (, ) Solutio Begi otig that is cotiuous ad oegative o the iterval 0, Net, partitio the iterval 0, ito suitervals, each o width Accordig to the deiitio o area, ou ca choose a -value i the ith suiterval For this eample, the right edpoits c i i are coveiet (0, 0) The area o the regio ouded the graph o, the -ais, 0, ad is Figure Editale Graph Area lim c i lim i lim i lim lim Right edpoits: c i i The area o the regio is Tr It EXAMPLE 6 Eploratio A Fidig Area the Limit Deiitio () = Fid the area o the regio ouded the graph o, the -ais, ad the vertical lies ad, as show i Figure 6 Solutio The uctio is cotiuous ad oegative o the iterval,, ad so egi partitioig the iterval ito suitervals, each o width Choosig the right edpoit c i a i i Right edpoits The area o the regio ouded the graph o, the -ais,, ad is Figure 6 Editale Graph o each suiterval, ou otai Area lim c i lim i lim i i lim i i lim 6 The area o the regio is Tr It Eploratio A Ope Eploratio

9 SECTION Area 67 The last eample i this sectio looks at a regio that is ouded the -ais (rather tha the -ais) EXAMPLE 7 A Regio Bouded the -ais Fid the area o the regio ouded the graph o 0, as show i Figure 7 ad the -ais or (, ) () = (0, 0) The area o the regio ouded the graph o ad the -ais or 0 is Figure 7 Editale Graph Solutio Whe is a cotiuous, oegative uctio o, ou still ca use the same asic procedure show i Eamples ad 6 Begi partitioig the iterval 0, ito suitervals, each o width The, usig the upper edpoits c i i, ou otai Area lim c i lim i The area o the regio is lim i lim 6 lim 6 Upper edpoits: c i i Tr It Eploratio A

10 SECTION Area 67 Eercises or Sectio The smol Click o Click o idicates a eercise i which ou are istructed to use graphig techolog or a smolic computer algera sstem to view the complete solutio o the eercise to prit a elarged cop o the graph I Eercises 6, id the sum Use the summatio capailities o a graphig utilit to veri our result i k c 6 I Eercises 7, use sigma otatio to write the sum k0 k 9 6 kk k j j i i 0 I Eercises 0, use the properties o summatio ad Theorem to evaluate the sum Use the summatio capailities o a graphig utilit to veri our result 0 i i 8 9 ii 0 I Eercises ad, use the summatio capailities o a graphig utilit to evaluate the sum The use the properties o summatio ad Theorem to veri the sum 0 i i i i 0 i 0 ii

11 68 CHAPTER Itegratio I Eercises 6, oud the area o the shaded regio approimatig the upper ad lower sums Use rectagles o width 6 I Eercises 7 0, use upper ad lower sums to approimate the area o the regio usig the give umer o suitervals (o equal width) I Eercises, id the limit o s as s 8 s 6 6 s s 8 I Eercises 8, use the summatio ormulas to rewrite the epressio without the summatio otatio Use the result to id the sum or 0, 00, 000, ad 0,000 i j 6 j 6kk i 7 8 i k I Eercises 9, id a ormula or the sum o terms Use the ormula to id the limit as 6i 9 lim 0 lim i lim i Numerical Reasoig Cosider a triagle o area ouded the graphs o, 0, ad (a) Sketch the regio () Divide the iterval 0, ito suitervals o equal width ad show that the edpoits are 0 < < < < (c) Show that s (d) Show that S (e) Complete the tale () Show that lim s lim S 6 Numerical Reasoig Cosider a trapezoid o area ouded the graphs o, 0,, ad (a) Sketch the regio () Divide the iterval, ito suitervals o equal width ad show that the edpoits are < < < < (c) Show that s (d) Show that S (e) Complete the tale i i i i () Show that lim s lim S s S s S lim i lim i lim i

12 SECTION Area 69 I Eercises 7 6, use the limit process to id the area o the regio etwee the graph o the uctio ad the -ais over the give iterval Sketch the regio 7, 0, 8,, 9, 0, 0, 0, 6,,,, 6, [,, 0,,, 6,, 0 I Eercises 7 6, use the limit process to id the area o the regio etwee the graph o the uctio ad the -ais over the give -iterval Sketch the regio 7, 0 8 g, 9, 0 60, 6 g, 6 h, I Eercises 6 66, use the Midpoit Rule Area with to approimate the area o the regio ouded the graph o the uctio ad the -ais over the give iterval 6, 0, 6 6 ta, 0, 66 Programmig Write a program or a graphig utilit to approimate areas usig the Midpoit Rule Assume that the uctio is positive over the give iterval ad the suitervals are o equal width I Eercises 67 70, use the program to approimate the area o the regio etwee the graph o the uctio ad the -ais over the give iterval, ad complete the tale Approimate Area 67, 0, 68 i, 0, si, 0, , 69 ta, 70 cos, 8, Writig Aout Cocepts, 6 0, Approimatio I Eercises 7 ad 7, determie which value est approimates the area o the regio etwee the -ais ad the graph o the uctio over the give iterval (Make our selectio o the asis o a sketch o the regio ad ot perormig calculatios) 7, 0, (a) () 6 (c) 0 (d) (e) 8 Writig Aout Cocepts (cotiued) 7 si 0,, (a) () (c) (d) 8 (e) 6 7 I our ow words ad usig appropriate igures, descrie the methods o upper sums ad lower sums i approimatig the area o a regio 7 Give the deiitio o the area o a regio i the plae 7 Graphical Reasoig Cosider the regio ouded the graphs o 0,, ad 0, as show i the igure To prit a elarged cop o the graph, select the MathGraph utto 8 6 8, (a) Redraw the igure, ad complete ad shade the rectagles represetig the lower sum whe Fid this lower sum () Redraw the igure, ad complete ad shade the rectagles represetig the upper sum whe Fid this upper sum (c) Redraw the igure, ad complete ad shade the rectagles whose heights are determied the uctioal values at the midpoit o each suiterval whe Fid this sum usig the Midpoit Rule (d) Veri the ollowig ormulas or approimatig the area o the regio usig suitervals o equal width Lower sum: s Upper sum: Midpoit Rule: S M i i i (e) Use a graphig utilit ad the ormulas i part (d) to complete the tale s S M

13 70 CHAPTER Itegratio () Eplai wh s icreases ad S decreases or icreasig values o, as show i the tale i part (e) 76 Mote Carlo Method The ollowig computer program approimates the area o the regio uder the graph o a mootoic uctio ad aove the -ais etwee a ad Ru the program or a 0 ad or several values o N Eplai wh the Mote Carlo Method works [Adaptatio o Mote Carlo Method program rom James M Scoers, Approimatio o Area Uder a Curve, MATHEMATICS TEACHER 77, o (Feruar 98) Copright 98 the Natioal Coucil o Teachers o Mathematics Reprited with permissio] 0 DEF FNF(X)=SIN(X) 0 A=0 0 B=π/ 0 PRINT Iput Numer o Radom Poits 0 INPUT N 60 N=0 70 IF FNF(A)>FNF(B) THEN YMAX=FNF(A) ELSE YMAX=FNF(B) 80 FOR I= TO N 90 X=A+(B-A)*RND() 00 Y=YMAX*RND() 0 IF Y>=FNF(X) THEN GOTO 0 0 N=N+ 0 NEXT I 0 AREA=(N/N)*(B-A)*YMAX 0 PRINT Approimate Area: ; AREA 60 END True or False? I Eercises 77 ad 78, determie whether the statemet is true or alse I it is alse, eplai wh or give a eample that shows it is alse 77 The sum o the irst positive itegers is 78 I is cotiuous ad oegative o a,, the the limits as o its lower sum s ad upper sum S oth eist ad are equal 79 Writig Use the igure to write a short paragraph eplaiig wh the ormula is valid or all positive itegers θ 80 Graphical Reasoig Cosider a -sided regular polgo iscried i a circle o radius r Joi the vertices o the polgo to the ceter o the circle, ormig cogruet triagles (see igure) (a) Determie the cetral agle i terms o () Show that the area o each triagle is r si (c) Let A e the sum o the areas o the triagles Fid lim A 8 Modelig Data The tale lists the measuremets o a lot ouded a stream ad two straight roads that meet at right agles, where ad are measured i eet (see igure) (a) Use the regressio capailities o a graphig utilit to id a model o the orm a c d () Use a graphig utilit to plot the data ad graph the model (c) Use the model i part (a) to estimate the area o the lot Road Figure or 8 Figure or 8 8 Buildig Blocks A child places cuic uildig locks i a row to orm the ase o a triagular desig (see igure) Each successive row cotais two ewer locks tha the precedig row Fid a ormula or the umer o locks used i the desig (Hit: The umer o uildig locks i the desig depeds o whether is odd or eve) 8 Prove each ormula mathematical iductio (You ma eed to review the method o proo iductio rom a precalculus tet) (a) i () i Stream Road is eve Putam Eam Challege Figure or 79 Figure or 80 8 A dart, throw at radom, hits a square target Assumig that a two parts o the target o equal area are equall likel to e hit, id the proailit that the poit hit is earer to the ceter tha to a edge Write our aswer i the orm a cd, where a,, c, ad d are positive itegers This prolem was composed the Committee o the Putam Prize Competitio The Mathematical Associatio o America All rights reserved