MA123, Chapter 9: Computing some integrals (pp )

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1 MA13, Chpter 9: Computig some itegrls (pp ) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how to use the bsic summtio formuls d the it rules you lered i this chpter to evlute some defiite itegrls. Assigmets: Assigmet 19 Assigmet 0 The rules d formuls give below llow us to compute firly esily Riem sums where the umber of subitervls is rther lrge. We c lso get compct d mgeble expressios for the sum so tht we c redily ivestigte wht hppes s pproches ifiity. Summtio rules: [sr 1 ] c = c [sr ] (c k ) = c k [sr 3 ] ( k ± b k ) = k ± b k Note: The summtios rules re othig but the usul rules of rithmetic rewritte i the Σ ottio. For exmple, [sr ] is othig but the distributive lw of rithmetic c 1 + c + + c = c( ) [sr 3 ] is othig but the commuttive lw of dditio ( 1 ± b 1 ) + ( ± b ) + + ( ± b ) = ( ) ± (b 1 + b + + b ) Summtio formuls: [sf 1 ] k = ( + 1) [sf ] k = ( + 1)( + 1) 6 Proof: I the cse of [sf 1 ], let S deote the sum of the itegers 1,,3,...,. Let us write this sum S twice: we first list the terms i the sum i icresig order wheres we list them i decresig order the secod time: S = S = If we ow dd the terms log the verticl colums, we obti S = ( + 1) + ( + 1) + + ( + 1) = ( + 1). }{{} times This gives our desired formul, oce we divide both sides of the bove equlity by. I the cse of [sf ], let S deote the sum of the itegers 1,,3,...,. The trick is to cosider the sum [(k + 1) 3 k 3 ]. O the oe hd, this ew sum collpses to ( /// ) + (33 \\\ ///) 3 + (4 xx 3 \\\) ( \\\ 3 xxxxxxxxx) ( 1) 3 + (( + 1) 3 \\\) 3 = ( + 1) = O the other hd, usig our summtio rules together with [sf 1 ] gives us [(k + 1) 3 k 3 ] = [3k + 3k + 1] = 3 k + 3 k + ( + 1) 1 = 3S ( + 1) Equtig the right hd sides of the bove idetities gives us: 3S = If we solve for S d properly fctor the terms, we obti our desired expressio. 1

2 More summtio rules: The ext formuls c be verified i sequetil order usig the sme type of trick used i the proof for [sf ]. The proofs get icresigly more tedious. [sf 3 ] k 3 = ( + 1) [sf 4 ] k 4 = ( + 1)( + 1)( ) 4 30 Exmple 1: Evlute the sum 9 (5k + 8). Exmple : Evlute the sum 8 (5k + 8k + 1). Exmple 3: Evlute the sum 1 k=7 (k + 1). 13

3 Exmple 4: Evlute the sum 100 ( + 5k). k=3 Exmple 5: Evlute the sum Exmple 6: Evlute the sum

4 Exmple 7: Evlute the sum Exmple 8: If we write (Hit: Substitute coveiet vlue of to help you evlute A.) k 4 = 4( + 1)( + 1)( ). Wht is the vlue of A? A Exmple 9: If (k k) = ( + 1)( 1), fid A. A 15

5 Limits t ifiity: We eed to be ble to evlute its of the form d q() re both polyomils i. E.g., how does p() q(), behve? where p() There is geerl priciple tht mkes computig these its esy. The ide is tht, for lrge vlues of, the term with the highest power of hs the most ifluece o the behvior of the polyomil. I other words, whe is very lrge, the term with the highest power domites the other terms. Theorem: Let p() d q() be polyomils. The p() q() = highest order term of p() highest order term of q(). Exmple 10: Fid the it s teds to ifiity Exmple 11: Fid the it s teds to ifiity. ( + 1) Exmple 1: Fid the it s teds to ifiity

6 Computig its of Riem sums: Let f be positive vlued fuctio defied o itervl [, b]. I Chpter 8 we strted studyig the problem of fidig the re of the regio i the xy-ple udereth the grph of the fuctio f d lyig bove the x-xis. We first prtitioed the itervl [,b] ito subitervls of legths x 1, x,..., x, respectively. For k = 1,..., we picked represettive poits p 1,p,...,p i ech of the subitervls i which [,b] hs bee prtitioed. We the formed the Riem sum f(p k ) x k. The defiite itegrl of f from to b ws defied s b f(x)dx = f(p k ) x k, if the it exists. Altertively, if we set P = mx 1 i { x i} we c write the bove it s b f(x)dx = P 0 f(p k ) x k. This mes tht we re tkig the it s the legth of the logest subitervl of the prtitio of [,b] is pprochig zero. As we observed i Chpter 8, it is computtiolly esier to prtitio the itervl [, b] ito subitervls of equl legth: x = (b )/. If we the use the right edpoits of this regulr prtitio we hve see tht: b f(x)dx = f( + k x) x. Defiite itegrls d res: y We stress gi the fct tht if the fuctio f tkes o oly positive vlues the the defiite itegrl is othig but the re of the regio udereth the grph of f, lyig bove the x-xis, d bouded by the verticl lies x = d x = b. Distce trveled by object: If the positive vlued fuctio uder cosidertio is the velocity v(t) of object t time t, the the re udereth the grph of the velocity fuctio d lyig bove the t-xis represets the totl distce trveled by the object from t = to t = b. 0 b x Wht if the fuctio tkes o lso egtive vlues? y If f hppes to tke o both positive d egtive vlues, the the Riem sum is the sum of the res of rectgles tht lie bove the x-xis d the egtives of the res of rectgles tht lie below the x-xis. Pssig to the it, we obti tht, i geerl, defiite itegrl c be iterpreted s differece of res: 0 b x b f(x)dx = [re of the regio(s) lyig bove the x-xis] [re of the regio(s) lyig below the x-xis] 17

7 Exmple 13: formuls to first simplify. Evlute the it s teds to ifiity. Note tht you will hve to use the summtio 1 k + 9 Exmple 14: formuls to first simplify. Evlute the it s teds to ifiity. Note tht you will hve to use the summtio 1 ( k 7 ) Exmple 15: The itegrl 5 0 x dx Wht vlue of A must pper i the sum? is computed s the it of the sum ( A k A ). 18

8 Exmple 16: The itegrl Wht vlue should be used s A? 10 7 x dx is computed s the it of the sum ( 3 A + k 3. ) Exmple 17: The it is obtied by pplyig the defiitio of the itegrl to Wht is the fuctio f(x)? 1 3 ( + k f(x)dx. ) Exmple 18: Evlute the it 4 ( k ) Hit: Wht it would you compute to evlute the re uder the curve y = 4 x for x betwee 0 d? Wht is this re i geometric terms? y 0 x 19

9 y 7 Exmple 19: Give tht the re of the ellipse 49x + y = 49 is 7π, evlute the itegrl x dx (Hit: Thik of the defiite itegrl s re.) 0 1 x Exmple 0: A cr is trvelig due est. Its velocity (i miles per hour) t time t hours is give by v(t) =.5t + 10t How fr did the cr trvel durig the first five hours of the trip? v t Riem sum with = 0 terms