Fall 2004 Math Integrals 6.1 Sigma Notation Mon, 15/Nov c 2004, Art Belmonte

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1 Fll Mth 6 Itegrls 6. Sigm Nottio Mo, /Nov c, Art Belmote Summr Sigm ottio For itegers m d rel umbers m, m+,...,, we write k = m + m k=m The left-hd side is shorthd for the fiite sum o right. The ide of summtio k tkes o iteger vlues from m to. Other idices m be used, such s i d j. Properties of fiite summtio Let c be costt. The c k = c k k=m k=m k + b k ) = k + b k k=m k=m k=m k b k ) = k b k k=m k=m k=m Prticulr sums ) = k= b) c = c k= c) k = k= d) k = e) f) + ) + )+) 6 k= ) k + ) = k= k = + )+) + ) k= Hd Emples Appl formuls from the Summr where pplicble. These problems re essetill eercises i ptter recogitio. As such, the re ripe for computer implemettio, s we ll see i the MATLAB emples! 68/ Write the sum We hve 68/8 6 i = Write the sum We hve 68/ + j = Write the sum 6 i= i epded form. i + i + = j= j i epded form. j = + + ) + + ) + + ). f i ) i i epded form. i= We hve f i ) i = f ) + f ) + + f ). i = 68/ Write the sum i sigm ottio. We hve k= k k +.

2 68/8 Write the sum i sigm ottio. This is sum of reciprocls of squres, 69/ 6 k= k. Write the sum ) i sigm ottio. This is emple of ltertig sum, 69/ Fid the vlue of the sum 6 j+. j= ) k k. k= We use brute force. 6 j+ = j= = = 76 69/6 Fid the vlue of the sum. i= We hve = = ) =. 69/ i= i= Fid the vlue of the sum i= ) i i. Appl the formuls. ) i i = i i i= i= i= i= ) +) +) [Stop; this is fie.] = = + + ) [Compre with MATLAB.] = + 69/ i ) ) ) i Fid the limit lim +. i= ) + First compute the sum. Note tht the ide of summtio is i. The letter is fied positive iteger. It is i tht vries! i ) ) ) i + i= = i + ) i i= = i + ) i i= i= = ) + ) + ) ) + ) = + + ) ) + + ) = + + ) ) + + = ) = Now tke the limit: lim ) =. i ) ) ) i Therefore, lim + =. i= If tht ws t world o hurt, I do t kow wht is! This is wh we we use computers: the re better t ptter recogitio th ou re. See the correspodig MATLAB emple.

3 69/c) Evlute the telescopig sum 99 i= i ). i + L = limits,, if) L = The omeclture mes tht the sum collpses. ) ) ) ) ) 99 i i= i+ = = = = 97 MATLAB Emples s69 [revisited] Fid the vlue of the sum i= ) i i. The MATLAB commd smsum smbolic summtio) mkes quick work of this oe! The swer grees with the oe we obtied b hd Stewrt 69/ sms i our sum = simplifsmsumiˆ - i -, i,, )); prettour sum) echo off; dir off s697 Prove the formul for the sum of fiite geometric series with first term d commo rtio r. i= r i = + r + r + +r = r ) r Stewrt 69/7: Sum of fiite geometric series sms i r GS = smsum * rˆi-), i,, ); GS = simplifgs); prettgs) echo off; dir off s698 r - ) r - echo off; dir off / + / - / - / Evlute i= i. s69 [revisited] i ) ) ) i Fid the limit lim +. i= A hlf pge of hd smbolic mipultio is reduced to oe lie of code. C ou s power tool? I kew ou could Stewrt 69/ sms i S = smsum/ * *i/)ˆ + **i/) ), i,, ); S = epds); pretts) Rewrite the sum s ) i, fiite geometric series with i= = dr =. The ppl the result from the precedig ) ) problem to obti the sum = 6 ) Stewrt 69/8: Sum of PARTICULAR fiite geometric series sms i S = simplif smsum / ˆi-), i,, ) ); pretts) echo off; dir off -)

4 s69 m Evlute i + j). i= j= This is double fiite sum Stewrt 69/: A fiite DOUBLE sum sms i j m S = smsum smsumi+j, j,, ), i,, m ); S = simplifs); pretts) echo off; dir off / m + / m + m

5 Fll Mth 6 Itegrls 6. Are Mo, /Nov c, Art Belmote Summr Let f be fuctio defied o I = [, b] with f oi. We seek the re of the regio R bouded bove b the curve = f ),belowbthe-is, o the left b the verticl lie =, d o the right b the verticl lie = b. Approimte the re b ddig up the res of rectgulr strips s follows. ) We hve P = m {,,, } =, the legth of the logest subitervl i the prtitio. b) The sum of the res of pproimtig rectgles is f ) i i i= =.7))+.7))+9.7))+.7)) =. c) Here is grph of f d the pproimtig rectgles. Stewrt 77/: Midpoit rule b b Split the itervl [, b]itosubitervls whose edpoits costitute prtitio P : = < < < < < =b. Let i [ ] i, i be i the ith subitervl d i = i i be the legth of this subitervl. We defie the orm of P b P = m i. Now let the umber of subitervls icrese i idefiitel while the orm of P shriks to. The re A of R is A = lim f ) i i, P i= provided tht this limit of the sum of the res of the rectgles formed b the prtitios eists. Hd Emples Appl formuls from the Sectio 6. Summr whe ecessr. 77/ Let f ) = 6,[,b]=[, ], P = {,,,, }, d i =midpoit. ) Fid P, the orm of P. b) Fid f ) i i, the sum of the res of pproimtig i= rectgles s give i the Summr. c) Sketch the grph of f d the pproimtig rectgles. 77/8 Let f ) = cos,[,b]= [, π ] {,P=, π 6, π },π,π,d i =left edpoit. ) Fid P, the orm of P. b) Fid f ) i i, the sum of the res of pproimtig i= rectgles s give i the Summr. c) Sketch the grph of f d the pproimtig rectgles. ) We hve P = m { π 6, π, π, π 6 } = π 6, the legth of the logest subitervl i the prtitio. b) The sum of the res of pproimtig rectgles is f ) i i i= = ) π ) 6 + ) π ) + ) π ) + ) π6 ) = ) π.789. c) Here is grph of f d the pproimtig rectgles.

6 Stewrt 77/8: Left sum MATLAB Emples s77 Let f ) =. 77/... Fid the ect re uder the curve = f ) = + d bove the -is betwee = db=. Use equl subitervls d k to be the right edpoit of the kth subitervl. Also sketch the regio. NOTE: For brevit, we ll write for k= ). Here is sketch of the regio whose re we seek. Stewrt 77/ The legth of ech subitervl is k = = b = = wheres the right edpoit of the kth subitervl is k k = + k = +. The sum of the res of the pproimtig rectgles is f k ) = f k ) [sice is costt] = ) + k ) + + k ) = ) + 6 k + 9 k k = 6 + = ) + = k) + 9 k ) + +) + 9+)+) ) 6 + ) + 9 ) + ) + ) = S Now let toobti A = lim S = = + = 7 = 7.. The re is 7. squre uits. Also see MATLAB emple.) ) ) Sketch the regio tht lies uder the curve = f ) bove the -is from = to=. b) Fid epressio for R, the sum of the res of the pproimtig rectgles, tkig k to be the right edpoit d usig subitervls of equl legth. c) Fid the umericl vlues of the pproimtig res R for =,,. d) Fid the ect re of the regio. A dir file t the ed shows ll computtios d plot commds. ) Here is sketch of the regio whose re we seek. Stewrt 77/ b) We hve R =. c) Here re vlues of R for the requested vlues of. R d) The ect re is lim = lim ) = Stewrt 77/: Are of regio s limit of sum of res of pproimtig rectgles b) sms k f = ilie.* -.ˆ, ); = ; b = ; d = b-)/; d = step size k = + k*d; right edpoit of kth subitervl R = smsumfk)*d, k,, ); right sum R = epdsimplifr)); prettr) c) N = [ ]; RSN = []; for m = N RSN = [RSN subsr,, m)]; echo off ed R

7 echo o d) A = limitr,, if) re A =.8.6 Stewrt 77/. echo off; dir off Stewrt 77/g: Sketch of regio f = ilie.* -.ˆ, ); = lispce, ); = f); f = [ ]; f = [ ]; fillf,f, m ); hold o plot[-..], [ ], k, LieWidth, ) plot[ ], [-..], k, LieWidth, ) grid o; is[ ]) lbel ); lbel ); title Stewrt 77/ ) echo off; dir off s77 [revisited] Fid the ect re uder the curve = f ) = + d bove the -is betwee = db=. Use equl subitervls d k to be the right edpoit of the kth subitervl. MATLAB s smsum commd rpidl ields the eedful Stewrt 77/: Are of regio s limit of sum of res of pproimtig rectgles sms k f = ilie.ˆ +.* -, ); = ; b = ; d = b-)/; d = step size k = + k*d; right edpoit of kth subitervl RS = smsumfk)*d, k,, ); right sum RS = epdsimplifrs)); prettrs) 6 7/ / ---- A = limitrs,, if) re A = 7/.. Here re vlues of R for the requested vlues of. R The ect re is lim R = Stewrt 77/: Are of regio s limit of sum of res of pproimtig rectgles b) sms k f = ilie si), ); = ; b = pi; d = b-)/; d = step size k = + k*d; right edpoit of kth subitervl R = smsumfk)*d, k,, ); right sum R = epdsimplifr)); prettr) pi pi si----) / pi \ cos----) - \ / c) N = [ ]; RSN = []; for m = N RSN = [RSN subsr,, m)]; echo off ed R echo o d) A = limitr,, if) re A = echo off; dir off echo off; dir off s77 Cosider the regio below the curve = f ) = si bove the -is betwee = d=π. Compute the sum of the res of pproimtig rectgles usig equl subitervls d right edpoits for =,,. Guess the ect vlue of the re.

8 Fll Mth 6 Itegrls 6. The Defiite Itegrl Fri, 9/Nov c, Art Belmote Summr Defiitios Let f be fuctio defied o I = [, b]. NOTE: I this sectio we remove the restrictio tht f o I.) Split the itervl [, b]itosubitervls whose edpoits costitute prtitio P : = < < < < < =b. Ofte the i re equll spced d we hve regulr prtitio.) Let i [ ] i, i be i the ith subitervl d i = i i be the legth of this subitervl. Recll tht the orm of P is defied b P = m i. Now let the umber of subitervls icrese idefiitel while the orm of P shriksto. The defiite itegrl of f from to b is defied b b f ) d = lim f ) i i, P i= provided the limit eists. Whe this occurs, f is sid to be itegrble o [, b].hereresometerms. The process of computig the vlue of itegrl is clled itegrtio. The smbol is clled itegrl sig. It ws itroduced b Leibiz, oe of the ivetors of Clculus i the 68s. The fuctio f ) is the itegrd. The umbers d b re clled limits of itegrtio; is the lower limit d b the upper limit. The sums i= f i ) i re clled Riem sums. The itegrl defied bove is kow s the Riem itegrl. Sufficiet coditios for defiite itegrl to eist If oe of these coditios holds, the f is itegrble o [, b]. f is cotiuous o [, b]. f is piecewise cotiuous o [, b]; i.e., f is cotiuous o [, b] ecept for fiite umber of jump discotiuities. f is mootoic o [, b]; i.e., icresig o [, b]or decresig o [, b]. Properties of the defiite itegrl Let c, m,dmbe costts d let f d g be itegrble o [, b], where b. The the followig properties hold.. b cd =cb ). b. b f ) + g) d = b f ) g) d = b. b cf)d = c b. b f)d f ) d = c f)d + b c f)d + b g)d f)d b g)d f)d 6. If f o[,b], the b f ) d. I this cse, the itegrl represets the re uder the curve = f ) d bove the -is betwee = d = b, s i Sectio If f g o [, b], the b f ) d b g)d. 8. If m f M o [, b], the b m b ) f ) d M b ). 9. b f ) d b f) d. Rules for pproimtig the defiite itegrl b f ) d These re ver es to implemet i MATLAB. Ech uses regulr prtitios with equl step size subitervl legth) h = b ). The Midpoit Rule is the most ccurte of these three rules. Left sum rule: L = h k= f + kh) Right sum rule: R = h k= f + kh) Midpoit Rule: M = h k= f + k + )h) Miscelleous defiitios If = b,the b If > b,the b f)d = f)d =. f)d = b f)d, provided the ltter eists s limit. Hd Emples Appl formuls from the Sectio 6. Summr whe ecessr. 86/9 Use the Midpoit Rule with = to pproimte d.

9 Here = db=. We hve h = b ) = ) = d + k + )h = k +, k =,,,,. Now f ) =, so Midpoit Rule: M = h f + k + )h) k= = f k + ) k= = f ) + f ) + f ) + f 7 ) + f 9 ) = = = 8 8 =.. 87/6 7 Compute 6 db tkig the limit of right sums. Here = db=7, h = b ) = 7 )) = 9 d + kh = + 9k,k=,...,.Now f)=6,so Right sum rule: R = h f + kh) k= = 9 ) 9k f k= = 9 )) 9k 6 k= = 9 8 ) k k= = 9 ) )) 8 k k= k= = 9 8 ) + ) = 9 9) = 9 9 From sketch we see tht the vlue of the itegrl is the sum of the res of rectgle d trigle: ) + ) = Stewrt 87/ 87/ Evlute d b iterpretig it i terms of re. From sketch we see tht the vlue of the itegrl is the re of semicirculr regio: π ) = π..... Stewrt 87/ 87/7 Evlute d b iterpretig it i terms of res. From sketch we iterpret the itegrl to be the sum of siged res. The positive res bove the -is ectl ccel out the egtive res below the -is. Hece d =. 7 As,wehveR = Thus 6 d=9... Stewrt 87/7 87/ Evlute + db iterpretig it i terms of res...

10 87/8 Evlute d b iterpretig it i terms of res. From sketch we see tht the vlue of the itegrl is the sum of the res of two trigles: ) + ) = 6 = Stewrt 87/8 87/6 8 Write the sum f ) d + f)d s sigle itegrl. Swp d combie. 8 8 f ) d + f)d = f)d 87/8 / Write the combitio 6 f ) d f)d + f)d s sigle itegrl. 87/ Epress the limit lim s defiite itegrl. + i/) i= Emie the pieces d flesh them out little. lim ) i= + + i We recogize this s the limit of the right sums of the itegrl + d. 87/ Use the properties of the defiite itegrl to evlute f ) d + f)d + f)d. Rerrge d combie. ) f ) d + f)d ) = f)d + f)d = f)d f)d = + f)d Rerrge d combie. 87/ = = = 6 f)d ) 6 f ) d + f)d + ) f)d + f)d 6 f)d + f)d f)d 6 + f)d Use the properties of the defiite itegrl to verif the iequlit π 6 without evlutig the itegrl. π/ si d π π/6 For π 6 π,wehvem= si = M. Applig Propert 8 ields π π ) 6 π/ or π π/ 6 si d π π/6. π/6 π si d 6) π

11 88/6 Use Propert 8 to estimte + d. The right sum uderestimtes the itegrl, the left sum overestimtes it, d the middle sum gives the best estimte. Stewrt 86/8: Right sum Stewrt 86/8: Left sum Stewrt 86/8: Midpoit Rule For, we hve m = + = M. Applig Propert 8 ields ) + d ) or + d 6. 88/6 Use the properties of the defiite itegrl, together with Eercise see MATLAB emples below), to prove the iequlit 6 + d. O the itervl [, ], we hve +, whece d + d [Propert 7] + d [Eercise ] 6 + d. MATLAB Emples s868 The tble below gives vlues of fuctio obtied from eperimet. Use them to estimte 6 f ) d usig three equl subitervls with ) right edpoits, b) left edpoits, d c) midpoits. SUPPLEMENT: If the fuctio f is kow to be decresig fuctio, c ou s whether our estimtes re less th or greter th the ect vlue of the itegrl? 6 f ) Here re right, left, d middle sums. A dir file t the ed shows ll computtios.) R L M Stewrt 86/8 = ; b = 6; = ; = : : 6 = 6 d = diff) d = R = [8.. -.]; L = [9. 8..]; M = [ ]; R = R * d R =. L = L * d L = 9.8 M = M * d M =.8 echo off; dir off s86 Use the Midpoit Rule with =,, to pproimte + d. Illustrte the cse for =. Here re the middle sums d plot, followed b dir file. M M M Stewrt 86/: Midpoit rule Stewrt 86/: Midpoit rule =,, ) formt log = ; b = ; f = ilie sqrt +.ˆ), ); plot[..], [ ], k ) -is grid o; hold o

12 plot[ ], [- ], k ) -is is[.. - ]) lbel ); lbel ) title Stewrt 86/: Midpoit rule ) = ; = lispce, b, +); subitervls, prt pts d = diff); legths of subitervls! = f:) +.*b-)/); s = ; Riem sum S = * d midpoit rule S =.867 plot[) )], [ f))], k, LieWidth, ) plot[+) +)], [ f+))], k, LieWidth, ) for k = : midpoit fuc vls fill[k), k+), k+), k)],... [,, k), k)],, LieWidth, ) echo off ed echo o = lispce, b); = f); plot,, LieWidth, ); = ; = lispce, b, +); subitervls, prt pts d = diff); legths of subitervls! = f:) +.*b-)/); s = ; Riem sum S = * d midpoit rule S = = ; = lispce, b, +); subitervls, prt pts d = diff); legths of subitervls! = f:) +.*b-)/); s = ; Riem sum S = * d midpoit rule S = formt short echo off; dir off Stewrt 87/ sms b k f = ilie.ˆ, ); d = b-)/; d = step size k = + k*d; right edpoit of kth subitervl RS = smsumfk)*d, k,, ); right sum RS = epdsimplifrs)); prettrs) b b b - / / / ---- b b - / + / b + / / b - / / / ---- I = limitrs,, if); pretti) echo off; dir off / b - / s87 b Prove tht d= b Stewrt 87/ sms b k f = ilie, ); d = b-)/; d = step size k = + k*d; right edpoit of kth subitervl RS = smsumfk)*d, k,, ); right sum RS = epdsimplifrs)); prettrs) b b / + / / b + / ---- I = limitrs,, if); pretti) echo off; dir off - / + / b s87 b Prove tht d = b.