General properties of definite integrals
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1 Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties of this itegrl tht will llow us to use it effectively lter. We ow hve forml defiitio of defiite itegrl tht theoreticlly llows us to compute res uder curve for cotiuous fuctios. But oly theoreticlly! Tht defiitio does ot tell us how to compute defiite itegrl, uless we wt to woder through some complicted, possily mid-ogglig limit computtios. Still, i the spirit of mthemticl iquiry, let us see wht properties we c extrct from this defiitio: mye some of these properties will tell us how to compute it efficietly. I wht follows, let us ssume tht from ovious, ut useful little fct. Proof f x is cotiuous o, Techicl fct f exists f ( x) dx 0. I this cse there is o re, sice we re tlkig out lie segmet.. We strt Aother wy to see this is tht the se of y theoreticl itervl we my cosider is 0. Next, otice tht the defiite itegrl is defied through lgeric formul d therefore the ssumptio tht the itegrd e positive is ot relly eeded. However, we do eed to e creful with the iterprettio of wht the itegrl of egtive fuctio represets. f x 0 o, Kot o your figer, the we c still defie: i lim f x x f ( x) dx i However, this itegrl, if it exists, is egtive, eig the limit of sum of egtive qutities. It therefore represets the egtive of the re ove the curve, s represeted i this grph: Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge
2 A f xdx Techicl fct The dditive property of defiite itegrls for their limits y f x is fuctio tht is cotiuous o,, d c, the: c f ( x) dx f ( x) dx f ( x) dx c Exmple: 4 x dx We c compute the vlue of this itegrl y usig pproprite geometric formule! Notice tht this itegrl represets the re of the regio shded i this grph. But this is trpezoid elow the x-xis, hece the itegrl will equl the egtive of its re. Therefore: 4 x dx B h How do we del with fuctios tht re sometimes positive d sometime egtive? To figure tht out, we otice the followig simple fct. Proof Well, this is oly ituitive sketch of the proof, ut the detils re ot difficult to check. f x 0 o,, the equtio simply sttes tht the re uder the curve from to c e cosidered s the sum of two res uder the curve, oe from to c, the other from c to. I the more geerl cse, we use the lgeric defiitio of the defiite itegrl s the limit of sum d otice tht the ove split is simply reflectig the split of the sum ito two sums, ech cotiig portio of the terms. I oth cses the coclusio is correct. This property llows us to uderstd the meig of y defiite itegrl i terms of res. Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge
3 Techicl fct y f x is y fuctio tht is cotiuous o,, we c still defie its defiite itegrl s: i f ( x) dx lim f x x the itegrl exists, it represets the differece etwee the re of the regio ouded y the curve ove the x xis d the re of the regio ouded y the curve elow the x xis: A i Proof Techicl fct y f x is fuctio tht is cotiuous o,, the: f ( x) dx f ( x) dx This fct is justified y oticig tht y goig i the reverse directio log the x xis, we re usig egtive widths for the rectgulr slices, so tht the differetils hve differet sigs o the two sides d therefore provide opposite vlues for the itegrls. A f ( x) dx A A Techicl fct f x d g x re cotiuous fuctios o,, the: f x g x dx f x dx g x dx The ext properties re lso firly ituitive d will prove very useful whe workig more itesely with defiite itegrls. Proof Here we just eed to rememer tht itegrl is limit of sum d tht Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge
4 sums crete o prolems whe computig limits. Therefore the two terms of the sum c e seprted ito to differet itegrls. I formul: lim i i f x g x dx f x g x x i lim f xi x g xi x i i lim lim f x x g x x i i i f x dx g x dx There is other property tht is very similr to the lst oe. Techicl fct f x is cotiuous o, umer, the: d k is y rel kf x dx k f x dx i I guess tht the proof here is very similr to the previous oe. Certily, d you will fid the chllege of developig i the lerig ctivities. The lst two fcts re remiiscet of the correspodig properties for idefiite itegrl. However, rememer tht so fr, tiderivtives re ot prt of ythig we hve doe! We re ow redy to move to the rel met of potto of this chpter, to the reso why clculus is such ig del i the history of sciece d techology. So, o to the ext two sectios! Wht, o exmples? There will e plety i those sectios. For ow, just reflect o the meig of the fcts we hve see i this sectio d check tht you uderstd such meigs through the lerig questios. Rememer tht we still do t kow how to compute y of these itegrls efficietly d elieve me you do t wt to compute y of them y usig the limit defiitio, just s you did ot like to do compute derivtives through the defiitio! OK, I look forwrd to wht is comig the. Summry A defiite itegrl my e defied for y fuctio, ot just positive oes, ut its iterprettio i terms of res must e modified. I geerl, the defiite itegrl of fuctio represets the differece etwee the res of the regios ouded y the fuctio ove the x xis d those ouded elow it. Defiite itegrls c e split log the limits, or log sum of the itegrd. Both properties ed up eig useful i pplictios. Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge 4
5 Commo errors to void Just ecuse the theoreticl fcts listed i this sectio seem resole, it does ot me tht they re cler to you! Chllege yourself with s my questios out them s possile d mke sure you c expli your rgumets d coclusios to yoe, eh? Lerig questios for Sectio I 4- Review questios:. Expli the reltioship etwee the defiite itegrl of cotiuous fuctio d the re of the regios such fuctio ouds.. Provide cler expltio of why defiite itegrls hve the dditio properties with respect to the itegrd d to the limits of itegrtio. Memory questios:. Whe does defiite itegrl represet re?. Stte the dditio property of the defiite itegrl with respect to the limits.. Stte the dditio property of the defiite itegrl with respect to the itegrd. 4. Wht hppes to defiite itegrl if we switch the limits of itegrtio? Computtio questios: I questios -4, use Riem itegrl ottio to express the exct vlue of the re of the regio ouded y the give curves... y x x y x x 6 8, 0, 4, 6 y x x y x x 6 8, 0,,. y cos x, y 0, x 0, x 4. y cos x, y 0, x, x Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge
6 I questios -0, express the give itegrl i terms of res of regios.. 4 x dx x dx 9. l xdx 6. 4 x dx x dx l xdx I questios - use sic geometric formule to determie the vlue of the give itegrl.. x dx x dx. f x is cotiuous fuctio for which 8 f ( x) dx 9 d 8 f ( x) dx, determie f ( x) dx d clerly expli how you oti your coclusio. 4. Use properties of defiite itegrls to determie the vlue of x x dx. Theory questios:. Whe you re sked to compute defiite itegrl with costt limits, should your coclusio cosist of costt, fuctio, fmily of fuctios or somethig else?. Preset defiite itegrl hvig y x s itegrd d hvig 0 s vlue.. How is the vlue of si x cos x dx relted to the vlue of si x cos x dx? Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge 6
7 4. Wht is the geometric iterprettio of the itegrl 0 cos xdx?. Wht is the geometricl iterprettio of the differetil dx i defiite itegrl? 6. I wht situtios is defiite itegrl egtive? 7. Does the itegrl x 4 0 dx represet the re of regio? 8. The re uder the curve x dx 0 y x etwee 0 d is give y the itegrl. Here x d dx re mesured i uits of legth d the itegrd is squre fuctio. So, why is the vlue of the itegrl re, mesured i squre uits isted of volume, mesured i cuic uits? 9. Wht is simple procedure to use whe computig the re of regio descried with x s fuctio of y? Proof questios:. Prove tht kdx k for y costts, d k.. Prove tht if f x is cotiuous o,, the kf ( x) dx k f ( x) dx Wht questios do you hve for your istructor? Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge 7
8 Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio : Geerl properties of defiite itegrls Pge 8
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