1.1 The FTC and Riemann Sums. An Application of Definite Integrals: Net Distance Travelled

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1 mth 3 more o the fudmetl theorem of clculus The FTC d Riem Sums A Applictio of Defiite Itegrls: Net Distce Trvelled I the ext few sectios (d the ext few chpters) we will see severl importt pplictios of defiite itegrls Whe first tkig clculus it is esy to cofuse the itegrtio (with its Riem sums) process with simple tidifferetitio While the First Fudmetl Theorem coects these two, they re ot the sme thig Most importt, though, the determitio of my qutities c e pproximted (iterpreted) s Riem sums d hece evluted s defiite itegrls eve though it is ot ovious t the outset tht tidifferetio should e ivolved The Riem sum prt turs out to e criticl Here s exmple of wht I me Suppose we kow tht the velocity of oject trvelig log lie (thik cr o stright highwy) is give y cotiuous fuctio v(t), where t represets time o the itervl [, ] How might we determie the et distce the oject hs trvelled? Well, we kow tht if the velocity were costt, the distce rte time Oserve: Distce hs ee expressed s product, much the wy we ssumed erlier tht the re of rectgle could e expressed s product: re of rectgle height se We c exted this logy to Riem sums d re uder curves While the velocity is ot costt o log itervls sice the velocity is cotiuous it is erly costt o short time itervls So divide the time itervl usig regulr prtitio {t o, t, t,,t } of suitervls of legth Dt Next, pick y poit i the kth suitervl (we might s well choose the right-hd edpoit t k for coveiece) d evlute the velocity v(t k ) there The the distce trveled durig the kth time itervl pproximted s distce rte time v(t k ) Dt Sice the et distce trvelled is the sum of the distces trveled o ech suitervl which is pproximtely Net Distce  v(t k ) Dt k The pproximtio is improved y lettig get lrge d tkig limit Net Distce lim!  k v(t k ) Dt v(t) dt (8) Sice v ws ssumed to e cotiuous, the y Theorem 3 we kow tht the limit exists d c e evluted s defiite itegrl usig tidifferetitio ssumig we kow pproprite tiderivtive Filly, thik out how we iterpreted defiite itegrls geometriclly: s (et) re uder curve Wht we hve just show is tht the et distce trvelled over the time itervl [, ] is just the et re uder the velocity curve Tht s ot ovious t first But eig le to

2 mth 3 more o the fudmetl theorem of clculus 3 Wht s your poit? The key poit here is tht we were le to use divide d coquer process to determie the displcemet Let s list it s series of steps We sudivided the qutity ito smll its, d we were le to pproximte the ech it s product Whe we ressemled (summed) the its, we foud we hd Riem sum Oce we hd Riem sum we could tke limit s the umer of its got lrge The limit ws defiite itegrl which we could evlute esily (if we kow tiderivtive) usig the First Fudmetl Theorem of Clculus We will use this process repetedly over the ext few weeks Look for it i other courses Wht qutities do you kow re products? Wht out the mout of electricity used i your home? If you kow the flow rte of electricity ito your house (go look t your electric meter spiig roud), the the mout of electricity cosumed c e computed s itegrl, just s we did with velocity (rte) d distce EXAMPLE Ok, we etter do oe exmple If the velocity of oject movig log stright lie is give y v(t) t + 3 si t m/s o the itervl [, p] Fid the et distce trvelled SOLUTION Ok, we just eed to use (3) Z p Net Distce v(t) dt t + 3 si tdt p t 3 cos t (p + 3) ( 3) p + 6 m YOU TRY IT If the velocity of oject movig log stright lie is give y v(t) t + p t m/s o the itervl [, 4] Fid the et distce trvelled (Aswer: l m)

3 mth 3 more o the fudmetl theorem of clculus 8 3 A Applictio of Defiite Itegrls: Averge Vlue Here is other simple exmple of pplictio of the defiite itegrl which poits out the power of the defiitio of the itegrl s Riem sum Suppose tht we wt to kow the verge temperture for Ferury 7, 3 i Geev (see Figure 49) How might we fid it? Well, we could tke the 4 hourly temperture recordigs, dd them together, d the divide y 4 might s we might do to fid y verge Is the verge 97 s listed i the tle? Wht verge is tht? Temp F Figure 44: A grph of the temperture o Ferury 7, 3 usig the dt to the right Time Temp : : 3 3: 3 4: 5: 6: 7: 3 8: 8 9: : 4 : 6 : 8 3: 8 4: 9 5: 9 6: 7 7: 5 8: 4 9: : 9 : 7 : 8 3: 7 4: 6 Ave 97 Time The verge of 97 privileges those recordigs mde o the hour We could get etter estimte if we recorded tempertures every hlf-hour, or every 5 miutes, or every miute, or perhps every secod The more recordigs we use, the etter the verge Let s geerlize the prolem I doig so, we re sometimes le to see the ptter which will help us solve the prticulr prolem we re iterested i The Averge Vlue Prolem: Let f e cotiuous fuctio o the closed itervl [, ] Fid the verge vlue of f o [, ] SOLUTION We mke use of the outlie of steps o pge 3 But how do we sudivide verge d mke it product? As usul, strt y dividig [, ] ito equl suitervls with prtitio poits {x, x,,x } The, s we suggested ove, Averge of f f (x )+ f (x )+ + f (x )  f (x k ) () k The summtio looks lmost like Riem sum except we ow hve isted of Dx But hold o! Dx so Dx Sustitutig this ck i equtio (4) gives Averge of f  f (x k ) k Dx  f (x k )Dx () k

4 mth 3 more o the fudmetl theorem of clculus 9 Now we do hve Riem sum i (5) We hve lredy remrked tht if we let icrese (tke more poits i our verge), we should get more ccurte pproximtio The est pproximtio occurs whe we tke limit s the umer of evlutio poits! I other words Averge of f lim! Â f (x k )Dx f (x) dx () k We kow this limit exists d equls the defiite itegrl ecuse f is cotiuous (see Theorem 3) Hvig crried out the steps o pge 3, we re led to mke the followig defiitio DEFINITION 6 (Averge Vlue) Assume tht f is itegrle o [, ] The the verge vlue of f o [, ] is deoted y f f ve d is defied y f f ve f (x) dx EXAMPLE 3 Fid the verge vlue of f (x) p x o [, 9] SOLUTION Usig Defiitio 7 f f ve f (x) dx 9 Z 9 p xdx x3/ The verge vlue is show i Figure 45 Thik of the origil curve s wve i fish tk The wve settles to the verge vlue of the fuctio The re of the rectgle formed usig the verge vlue s the height is the sme s the re uder the origil curve Notice tht there is poit, mely c 4 t which the height of the curve is the sme s the verge vlue ( f (4) ) 7 (7 ) c Figure 45: The verge vlue of p x o [, 9] is This vlue ctully occurs t c 4 YOU TRY IT 6 Fid the verge vlue of f (x) x o [ itervl does the verge vlue ctully occur?, ] For which vlues c i the Aswer to you try it 6 : The verge vlue is It occurs t c ± EXAMPLE 3 A ptiet eig treted for emphysem is tested with spirometer to mesure lug cpcity The dt show the volume of ir i the ptiet s lugs durig ihltio is give y V(t) cos pt pits over the time itervl [, ] secods Fid the verge volume of ir i the his lugs durig this period SOLUTION Usig Defiitio 7 Averge V Z pt cos dt pple pt t p si [( ) ( )] pit I Figure 46 otice how the re uder the curve ove the verge vlue lces out the missig re elow the verge vlue d the curve Notice lso tht the verge vlue ctully occurs t c (sice f () ) Figure 46: The verge vlue of V(t) cos pt o [, ] is YOU TRY IT 7 A ptiet eig treted for pulmory firosis is tested with spirometer to mesure lug cpcity The dt show the volume of ir i the ptiet s lugs durig oth the ihltio d exhltio cycles is give y pt V(t) cos pits 5 over the time itervl [, 5] secods Fid the verge volume of ir i the his lugs durig this period At wht time(s) does this volume occur? Aswer to you try it 7 : pit d it occurs t c d 4

5 mth 3 more o the fudmetl theorem of clculus 3 Moolight Hrs Figure 47: Dt (for Bosto) from The Old Frmer s Almc for the umer of hours the moo is visile per dy durig its 8-dy cycle Dy YOU TRY IT 8 (Hd i: Chllege) The phses of the moo occur i regulr, predictle 8 dy lur cycle Dt (for Bosto) from The Old Frmer s Almc idicte tht the hours, v(x), the moo is visile per dy over the course of the cycle is v(x) 65 cos( 4 p x p)+ 435, where x is the dy of the cycle () Fid the verge umer of hours the moo is visile per dy i Bosto usig clculus () Extr Credit: Use your ri to expli why you should hve expected this swer! I Exmple 3 d Exmple 3 we oted tht there were poits where the verge vlue of the fuctio ctully occurred This turs out to lwys e true s log s f (x) is cotiuous THEOREM (Me Vlue Theorem for Itegrls: MVTI) If f is cotiuous o [, ], the there s poit c i [, ] so tht I other words f (c) f (t) dt f (c) ( ) f (t) dt f f ve Proof As usul, for x i [, ], defie our ccumultio fuctio s A(x) Z x f (t) dt The y FTC I, A (x) f (x) SoA(x) is differetile o [, ] so it is cotiuous there By the origil MVT, there is poit c i [, ] so tht A (c) A() A() R f (t) dt R R f (t) dt f (t) dt f (t) dt But A (c) f (c), so f (c) f (t) dt f f ve EXAMPLE 33 Let f (x) cos(x) o p, p Fid the poit c i f ve p, p where f (c) SOLUTION First otice tht cos x is differetile so it is cotiuous So the MVTI pplies Next we eed to determie f f ve Usig Defiitio 7 f f ve Z p/ cos xdx p/ p/ ( p/) p/ p ( si x) p/ p [ ( )] 4 p

6 mth 3 more o the fudmetl theorem of clculus 3 So we eed to fid c i p, p so tht f (c) f fve p 4 I other words, f (c) cos c 4 p cos c p c rccos ± p YOU TRY IT 9 Let f (x) x o [, 3] Does the MVTI pply to this fuctio? Why? If so, fid the poit(s) c i [, 3] so tht f (c) f f ve EXAMPLE 34 Fid the verge vlue of f (x) x 3 o [, 5] d determie the poits c where the verge occurs Aswer to you try it 9 : The MVTI does pply ecuse f is polyomil (cotiuous) c p 3 SOLUTION By Defiitio 7, f f ve f (x) dx Z 5 5 See grph 3 x 3 dx pple (3)(3)+ ()() Note tht we were le to esily evlute the itegrl y usig the geometry of the the two trigles Sice x 3 is cotiuous, the MVTI sys there s poit c i [, 5] where f (c) f f ve We eed 5 3 f (x) x Figure 48: The re uder f (x) x 3 o [, 5] cosists of trigles c 3 3 () ( c 3 3 c 3 3 () ( c 43 c 7 Both poits re i the itervl