Riemann Sums and Definite Integrals. Riemann Sums

Transcription

1 _.qd // : PM Pge 7 SECTION. Riem Sums d Defiite Itegrls 7 Sectio. Riem Sums d Defiite Itegrls Uderstd the defiitio of Riem sum. Evlute defiite itegrl usig its. Evlute defiite itegrl usig properties of defiite itegrls. Riem Sums I the defiitio of re give i Sectio., the prtitios hve suitervls of equl width. This ws doe ol for computtiol coveiece. The followig emple shows tht it is ot ecessr to hve suitervls of equl width. EXAMPLE A Prtitio with Suitervls of Uequl Widths... f() =... ( ) The suitervls do ot hve equl widths. Figure.8 = (, ) Are = (, ) The re of the regio ouded the grph of d the -is for is. Figure.9 Cosider the regio ouded the grph of f d the -is for, s show i Figure.8. Evlute the it f c i i i where c is the right edpoit of the prtitio give c i i i d i is the width of the ith itervl. Solutio So, the it is The width of the ith itervl is give i i i i i i i. f c i i i i i. i i i i From Emple 7 i Sectio., ou kow tht the regio show i Figure.9 hs re of. Becuse the squre ouded d hs re of, ou c coclude tht the re of the regio show i Figure.8 hs re of. This grees with the it foud i Emple, eve though tht emple used prtitio hvig suitervls of uequl widths. The reso this prticulr prtitio gve the proper re is tht s icreses, the width of the lrgest suitervl pproches zero. This is ke feture of the developmet of defiite itegrls.

2 _.qd // : PM Pge 7 7 CHAPTER Itegrtio The Grger Collectio GEORG FRIEDRICH BERNHARD RIEMANN (8 8) Germ mthemtici Riem did his most fmous work i the res of o-euclide geometr, differetil equtios, d umer theor. It ws Riem s results i phsics d mthemtics tht formed the structure o which Eistei s theor of geerl reltivit is sed. I the precedig sectio, the it of sum ws used to defie the re of regio i the ple. Fidig re this mes is ol oe of m pplictios ivolvig the it of sum. A similr pproch c e used to determie qutities s diverse s rc legths, verge vlues, cetroids, volumes, work, d surfce res. The followig defiitio is med fter Georg Friedrich Berhrd Riem. Although the defiite itegrl hd ee defied d used log efore the time of Riem, he geerlized the cocept to cover roder ctegor of fuctios. I the followig defiitio of Riem sum, ote tht the fuctio f hs o restrictios other th eig defied o the itervl,. (I the precedig sectio, the fuctio f ws ssumed to e cotiuous d oegtive ecuse we were delig with the re uder curve.) Defiitio of Riem Sum Let f e defied o the closed itervl,, d let e prtitio of, give < < <... < < where i is the width of the ith suitervl. If c i is poit i the ith suitervl, the the sum f c i i, i i c i i is clled Riem sum of f for the prtitio. NOTE The sums i Sectio. re emples of Riem sums, ut there re more geerl Riem sums th those covered there. The width of the lrgest suitervl of prtitio is the orm of the prtitio d is deoted. If ever suitervl is of equl width, the prtitio is regulr d the orm is deoted. Regulr prtitio 8 = does ot impl tht. Figure. For geerl prtitio, the orm is relted to the umer of suitervls of, i the followig w. Geerl prtitio So, the umer of suitervls i prtitio pproches ifiit s the orm of the prtitio pproches. Tht is, implies tht. The coverse of this sttemet is ot true. For emple, let e the prtitio of the itervl, give < < <... < 8 < < <. As show i Figure., for positive vlue of, the orm of the prtitio is. So, lettig pproch ifiit does ot force to pproch. I regulr prtitio, however, the sttemets d re equivlet.

3 _.qd // : PM Pge 7 SECTION. Riem Sums d Defiite Itegrls 7 Defiite Itegrls To defie the defiite itegrl, cosider the followig it. f c i i L i To s tht this it eists mes tht for > there eists > such tht for ever prtitio with < it follows tht L i f c i <. i (This must e true for choice of i the ith suitervl of. ) c i FOR FURTHER INFORMATION For isight ito the histor of the defiite itegrl, see the rticle The Evolutio of Itegrtio A. Sheitzer d J. Steprs i The Americ Mthemticl Mothl. To view this rticle, go to the wesite Defiitio of Defiite Itegrl If f is defied o the closed itervl, d the it f c i i i eists (s descried ove), the f is itegrle o, d the it is deoted f c i i f d. i The it is clled the defiite itegrl of f from to. The umer is the lower it of itegrtio, d the umer is the upper it of itegrtio. It is ot coicidece tht the ottio for defiite itegrls is similr to tht used for idefiite itegrls. You will see wh i the et sectio whe the Fudmetl Theorem of Clculus is itroduced. For ow it is importt to see tht defiite itegrls d idefiite itegrls re differet idetities. A defiite itegrl is umer, wheres idefiite itegrl is fmil of fuctios. A sufficiet coditio for fuctio f to e itegrle o, is tht it is cotiuous o,. A proof of this theorem is eod the scope of this tet. THEOREM. Cotiuit Implies Itegrilit If fuctio f is cotiuous o the closed itervl,, the f is itegrle o,. EXPLORATION The Coverse of Theorem. Is the coverse of Theorem. true? Tht is, if fuctio is itegrle, does it hve to e cotiuous? Epli our resoig d give emples. Descrie the reltioships mog cotiuit, differetiilit, d itegrilit. Which is the strogest coditio? Which is the wekest? Which coditios impl other coditios?

4 _.qd // : PM Pge 7 7 CHAPTER Itegrtio EXAMPLE Evlutig Def iite Itegrl s Limit Evlute the defiite itegrl d. f() = Becuse the defiite itegrl is egtive, it does ot represet the re of the regio. Figure. Solutio The fuctio f is itegrle o the itervl, ecuse it is cotiuous o,. Moreover, the defiitio of itegrilit implies tht prtitio whose orm pproches c e used to determie the it. For computtiol coveiece, defie sudividig, ito suitervls of equl width i Choosig c i s the right edpoit of ech suitervl produces c i i i. So, the defiite itegrl is give d f c i i i f c i i i i i i f Becuse the defiite itegrl i Emple is egtive, it does ot represet the re of the regio show i Figure.. Defiite itegrls c e positive, egtive, or zero. For defiite itegrl to e iterpreted s re (s defied i Sectio.), the fuctio f must e cotiuous d oegtive o,, s stted i the followig theorem. (The proof of this theorem is strightforwrd ou simpl use the defiitio of re give i Sectio..) You c use defiite itegrl to fid the re of the regio ouded the grph of f, the -is,, d. Figure. THEOREM. The Defiite Itegrl s the Are of Regio If f is cotiuous d oegtive o the closed itervl,, the the re of the regio ouded the grph of f, the -is, d the verticl lies d is give Are f d. (See Figure..)

5 _.qd // : PM Pge 7 SECTION. Riem Sums d Defiite Itegrls 7 f() = Are d Figure. As emple of Theorem., cosider the regio ouded the grph of d the -is, s show i Figure.. Becuse f is cotiuous d oegtive o the closed itervl,, the re of the regio is Are d. A strightforwrd techique for evlutig defiite itegrl such s this will e discussed i Sectio.. For ow, however, ou c evlute defiite itegrl i two ws ou c use the it defiitio or ou c check to see whether the defiite itegrl represets the re of commo geometric regio such s rectgle, trigle, or semicircle. EXAMPLE Ares of Commo Geometric Figures Sketch the regio correspodig to ech defiite itegrl. The evlute ech itegrl usig geometric formul. f. d. d c. d NOTE The vrile of itegrtio i defiite itegrl is sometimes clled dumm vrile ecuse it c e replced other vrile without chgig the vlue of the itegrl. For istce, the defiite itegrls d d t dt hve the sme vlue. Solutio A sketch of ech regio is show i Figure... This regio is rectgle of height d width. d (Are of rectgle) 8. This regio is trpezoid with ltitude of d prllel ses of legths d. The formul for the re of trpezoid is h. (Are of trpezoid) d c. This regio is semicircle of rdius. The formul for the re of semicircle is r. (Are of semicircle) d f() = () Figure. () f() = + f() =

6 _.qd // : PM Pge 7 7 CHAPTER Itegrtio Properties of Defiite Itegrls The defiitio of the defiite itegrl of f o the itervl, specifies tht <. Now, however, it is coveiet to eted the defiitio to cover cses i which or >. Geometricll, the followig two defiitios seem resole. For istce, it mkes sese to defie the re of regio of zero width d fiite height to e. Defiitios of Two Specil Defiite Itegrls. If f is defied t, the we defie f d.. If f is itegrle o,, the we defie f d f d. EXAMPLE Evlutig Defiite Itegrls. Becuse the sie fuctio is defied t, d the upper d lower its of itegrtio re equl, ou c write si d.. The itegrl d is the sme s tht give i Emple () ecept tht the upper d lower its re iterchged. Becuse the itegrl i Emple () hs vlue of, ou c write d d. f f() d I Figure., the lrger regio c e divided t c ito two suregios whose itersectio is lie segmet. Becuse the lie segmet hs zero re, it follows tht the re of the lrger regio is equl to the sum of the res of the two smller regios. c c f() d + Figure. c f() d THEOREM. Additive Itervl Propert If f is itegrle o the three closed itervls determied,, d c, the c f d f d f d. c EXAMPLE d Usig the Additive Itervl Propert d d Theorem. Are of trigle

7 _.qd // : PM Pge 77 SECTION. Riem Sums d Defiite Itegrls 77 Becuse the defiite itegrl is defied s the it of sum, it iherits the properties of summtio give t the top of pge. THEOREM.7 Properties of Defiite Itegrls If f d g re itegrle o, d k is costt, the the fuctios of kf d f ± g re itegrle o,, d. kf d k f d. f ± g d f d ± g d. Note tht Propert of Theorem.7 c e eteded to cover fiite umer of fuctios. For emple, f g h d f d g d h( d. g f EXAMPLE Evlute Solutio d, Evlutio of Defiite Itegrl d d, d usig ech of the followig vlues. d d d d d d d If f d g re cotiuous o the closed itervl, d f g f d g d Figure. for, the followig properties re true. First, the re of the regio ouded the grph of f d the -is (etwee d ) must e oegtive. Secod, this re must e less th or equl to the re of the regio ouded the grph of g d the -is (etwee d ), s show i Figure.. These two results re geerlized i Theorem.8. (A proof of this theorem is give i Appedi A.)

8 _.qd // : PM Pge CHAPTER Itegrtio THEOREM.8 Preservtio of Iequlit. If f is itegrle d oegtive o the closed itervl,, the f d.. If f d g re itegrle o the closed itervl, d f g for ever i,, the f d g d. I Eercises d, use Emple s model to evlute the it over the regio ouded the grphs of the equtios. I Eercises 8, evlute the defiite itegrl the it defiitio.. d.. d. 7. d 8. I Eercises 9, write the it s defiite itegrl o the itervl [, ], where is poit i the ith suitervl Eercises for Sectio. fc i i i. f,,, (Hit: Let c i i.). f,,, (Hit: Let c i i.) Limit c i i i c i c i i i c i i i i c i i c i d d d Itervl,,,, I Eercises, set up defiite itegrl tht ields the re of the regio. (Do ot evlute the itegrl.). f. f See for worked-out solutios to odd-umered eercises. f f 8. f f

9 _.qd // : PM Pge 79 SECTION. Riem Sums d Defiite Itegrls f si. f t I Eercises, sketch the regio whose re is give the defiite itegrl. The use geometric formul to evlute the itegrl >, r >.. d.. d. 7. d d.. 9 d. I Eercises, evlute the itegrl usig the followig vlues d, d d d d π 8 d d π. g. f 8 d, 8 r r d d 8 d d π π d r d 9... Give f d d f d, evlute (). Give f d d f d, evlute (). Give f d d g d, evlute (). Give f d d f d, evlute () f d. () f d f d. d d 7 f d. f d. f d. f d. f g d. g d. f d. 7. Use the tle of vlues to fid lower d upper estimtes of f d. () () () Assume tht f is decresig fuctio. f. Use the tle of vlues to estimte f d. Use three equl suitervls d the () left edpoits, () right edpoits, d midpoits. If f is icresig fuctio, how does ech estimte compre with the ctul vlue? Epli our resoig. f f d. f d. f d. f d. g f d. f d. f d

10 _.qd // : PM Pge 8 8 CHAPTER Itegrtio 7. Thik Aout It The grph of f cosists of lie segmets d semicircle, s show i the figure. Evlute ech defiite itegrl usig geometric formuls. () (e) 8. Thik Aout It The grph of f cosists of lie segmets, s show i the figure. Evlute ech defiite itegrl usig geometric formuls. () (e) 9. Thik Aout It Cosider the fuctio f tht is cotiuous o the itervl, d for which f d. Evlute ech itegrl. () () 7 f d ( f is eve.) f d ( f is odd.) f d f d f d f d f d f d (, ) f d f d (, ) () (f) () (f) f d f d f d (, ) 8 (8, ) f (, ) (, ) f f d f d f d. Thik Aout It A fuctio f is defied elow. Use geometric formuls to fid 8 f d. I Eercises 8, determie which vlue est pproimtes the defiite itegrl. Mke our selectio o the sis of sketch.. d () () (e) 8. cos d () () (e) 7. si d () () 8. f,, 9 d < Writig Aout Cocepts I Eercises d, use the figure to fill i the lk with the smol <, >, or.. The itervl, is prtitioed ito suitervls of equl width, d i is the left edpoit of the ith suitervl. f i f d i. The itervl, is prtitioed ito suitervls of equl width, d i is the right edpoit of the ith suitervl. f i f d i. Determie whether the fuctio f is itegrle o the itervl,. Epli.. Give emple of fuctio tht is itegrle o the itervl,, ut ot cotiuous o,. () () 9 7

11 _.qd // : PM Pge 8 SECTION. Riem Sums d Defiite Itegrls 8 Progrmmig Write progrm for our grphig utilit to pproimte defiite itegrl usig the Riem sum fc i i i where the suitervls re of equl width. The output should give three pproimtios of the itegrl where c i is the lefthd edpoit L, midpoit M, d right-hd edpoit R of ech suitervl. I Eercises 9, use the progrm to pproimte the defiite itegrl d complete the tle. 9. d.. si d. True or Flse? I Eercises 8, determie whether the sttemet is true or flse. If it is flse, epli wh or give emple tht shows it is flse... L M R. If the orm of prtitio pproches zero, the the umer of suitervls pproches ifiit.. If f is icresig o,, the the miimum vlue of f o, is f. 7. The vlue of f d must e positive. 8. The vlue of si d is. 8 f g d fg d 9. Fid the Riem sum for f f d over the itervl, 8, where,,, 7, d 8, d where c, c, c, d c 8. d si d f d g d g d 8 8 Figure for 9 Figure for 7 7. Fid the Riem sum for f si over the itervl,, where,,,, d, d where c, c, c, d c. 7. Prove tht d. 7. Prove tht d. 7. Thik Aout It Determie whether the Dirichlet fuctio f,, is itegrle o the itervl,. Epli. 7. Suppose the fuctio f is defied o,, s show i the figure. f,, Show tht f d does ot eist. Wh does t this cotrdict Theorem.? 7. Fid the costts d tht mimize the vlue of d. Epli our resoig. 7. Evlute, if possile, the itegrl 77. Determie is rtiol is irrtiol < d. π π usig pproprite Riem sum.