Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral

Transcription

1 Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ i i i= where i ξi i, Δ i = i i, i =,,...,. = f = ξ ξ =. Defiite Itegrl The limit of the sum S whe the umer of the suitervl pprohes ifiit tht the lrgest Δ pprohes zero is lle efiite itegrl of the futio f i with the upper limit = lower limit =. ( ξ ) lim f i Δ i = f m Δi i = or equivletl lim f ( ξi) Δ i = f i = If the futio f is otiuous o [, ] e itegrle o[, ]. or if the limit eists, the futio is si to If is i the omi of f, we efie f = If f is itegrle o[, ] f f. efie =, the we Emple Fi the Riem Sum S for the futio f = + over the itervl [,] iviig ito equl suitervls, the fi the limit lim S. Solutio 9 9i Δ i = = ξ i = i = + iδ i = +

2 Defiite Itegrl 9i 9i ξ i = + + = + S = f ξ Δ hee f the i i i= 9i 9 = + i= 8 8 = + i i= i= 8 8 = ( ) = = 8 + ( ) lim S = lim 8 + = 8 + = Emple ( ) 8 Solutio Divie the itervl [, ] ito equl suitervls. Hee we oti Δ i =. I eh suitervl[, i i i], hoose ξi suh tht ξ i = +Δ i i = + i f ( ξi) Δ i = + 8 i= i= 8i 6i = + 8 i= 6i i = 6 + i= 6i 8i = + i= 6 8 = ( ) + ( ) 6 ( + ) 8 ( + )( + ) = =

3 Defiite Itegrl ( 8) lim ( ξi) = f Δ i = i 8 = lim = + = Suitervl propert If f is itergrle o itervl otiig the poits,,, the = + f f f o mtter wht the orer of,,.. The first Fumetl Theorem of Clulus Theorem A First Fumetl theorem of Clulus, let e vrile poit i (, ), Let f e otiuous o the lose itervl [ ] the () = f t t f = f t f Proof For (, ) we efie F () = F( + h) F f t t F = lim h h + h = lim f t t h h + h = lim f () t t h h But + h f () t t = f t t, the () () f t t represets the re oue -is the urve + h + h, whih is pproimte to f () tt lim hf f = =. h h + h hf ; tht is f () t t hf f t etwee. So,

4 Defiite Itegrl t Emple t = t Emple t t ostt = t t ostt = t tostt = t os Emple Fi ( t ) t Solutio Let u = u = hee u ( t ) t ( t ) t = u u = ( t ) t u = u = 6 Theorem B Compriso Propert, If f g re itegrle o [ ] if f g for ll i [, ] f ( ) g( ) M m, the Proof Over the itervl[, ], let there e ritrr prtitio = < < < =. Let ξi e smple poit o the i th suitervl[, i i], the we olue tht f ( ξi) g( ξi) f ξ Δ g ξ Δ ( ξ ) ( ξ ) i i i i i= i= lim i i i i f Δ g Δ ( ξ ) lim ( ξ ) f Δ g Δ i i i i i= i= f g Theorem C Boueess Propert If f is itegrle o [, ] m f M for ll i[, ], the Proof Let h m, [, ] Hee, h f ( ) ( ) m f M =, the h f, [, ] ( ) m f. = f

5 Defiite Itegrl B similr w, let g M, [, ] f g, [, ] f ( ) g f M ( ) Therefore m ( ) f( ) M( ) =, the. Seo Fumetl Theorem of Clulus Me vlue theorem For Itegrls Seo Fumetl Theorem of Clulus Let f e itegrle o [, ] F e primitive of f o[, ], the f ( ) = F ( ) F ( ) It is lso kow s Newto-Leiiz Formul. For oveiee we itroue speil smol for F F writig F F = F or F ( ) F ( ) = F ( ) Emple Emple = = = = si os = = 9 si 8 8 Me Vlue Theorem for Itegrl If f is otiuous o[, ], there is umer etwee suh tht Proof Let, () = ( ) f tt f F = f t t B Me vlue theorem for erivtive, we oti F F = F () = ( ) f t t f () = ( ) f t t f = f () t t is lle the me vlue, or verge vlue of f o [, ] f Emple Fi the verge vlue of f = o the itervl [, ] Solutio 7 f = ve f = = = 5

6 Defiite Itegrl Emple Fi the verge vlue of f = os o the itervl [, ] 5. Chge of vrile i efiite itegrl If f is otiuous over the lose itervl, if = ϕ ( t) is otiuous its erivtive is ϕ ( t) over the itervlα t β, where = ϕ ( α ) = ϕ ( β ) if f ϕ ( t) is efie et otiuous over the itervl α t β, the β f = f ϕ( t) ϕ ( t) t α Emple Fi ( > ) Solutio Let = si t, = ost, t = rsi, α = rsi = β = rsi =. the we oti = si t si t os tt Emple Evlute Emple Evlute = si tos tt = si tt = = = ( ostt ) sit let = t (swer: l + ) l e let e = z (swer: 6. Itegrtio prts If the futios u = ) v re otiuous ifferetile over [, ] u v u v v u Emple Evlute os (swer: ) e + Emple Evlute e (swer: ) 8 Emple Evlute e si (swer: ( e + )), we hve 7. Improper Itegrl Improper itegrls refer to those ivolvig i the se where the itervl of itegrtio is ifiite lso i the se where f (the itegr) is uoue t fiite umer of poits o the itervl of itegrtio. 7. Improper Itegrl with Ifiite Limits of Itegrtio Let e fie umer ssume tht f N eists for ll N. The if 6

7 Defiite Itegrl N f eists, we efie the improper itegrl f lim N = lim N N f f The improper itegrl is si to e overget if this limit is fiite umer to e iverget otherwise. Emple Evlute I = Solutio N N = lim lim lim N = = + = N N N Thus, the improper itegrl overges hs the vlue. Emple Evlute p e Let e fie umer ssume f lim t t f eists we efie the improper itegrl The improper itegrl f eists for llt <. The if t = lim f f t t is si to e overge if this limit is fiite umer to iverge otherwise. If oth f A f overge for some umer, the improper itegrl of f o the etire -is is efie = + f f f Emple Evlute + (swer: ) (swer: ) Improper Itegrls with Uoue Itegrs If f is uoue t f t eists for ll t suh tht < t, the = lim f f + t t 7

8 Defiite Itegrl If the limit eists (s fiite umer), we s tht the improper itegrl overge; otherwise, the improper itegrl iverges. Similrl, if f is uoue t t f eists for ll t suh tht t <, the If f is uoue t where = lim f f t t oth overge, the = + < < the improper itegrl f f f f f = f = g A We s tht the itegrl o the left iverges if either or oth of the itegrls o the right iverge. Emple Fi ( ) Note:. For f g g f, if if overges, the f g overge Emple Ivestigte the overgee of iverges.. For ( + e ), if f g if f iverges, the Emple Ivestigte the overgee of +. If f is overget the f solute overget. si Emple Ivestigte the overgee of 8 Are Betwee Two Curves 8. Are Betwee = f = g If f g re otiuous futios o the itervl[, ], if f g for ll i [, ] the regio oue ove = f, elow = g, o the left lie, the the re of =, o the right the lie = is efie A= f g g is lso overget, speifill 8

9 Defiite Itegrl Emple Fi the re of regio oue ove = + 6, oue elow =, oue o the sies the lies = =. s: Emple Fi the re of the regio elose etwee the urves = = 8. Are Betwee = v( ) = w( ) If w v re otiuous futios if w( ) v( ) for ll i [, ], the the re of the regio oue o the left = v( ), o the right = w( ), elow =, ove = is efie Emple Fi the re of the regio elose with respet to. (s: 9 ) A = w v = v( ) w( ) =, itegrtig Emple Fi the re of the regio elose the urves itegrtig /. with respet to /. with respet to = = 8. Are i Polr Coorites A θ = β θ = α β r = ρ θ A = ρ θ α θ r = ρ = ( θ ) r ρ θ A θ = β θ = α β ( ) A = ρ θ ρ θ θ α Emple Clulte the re elose the rioi r = osθ (swer: ) Emple Fi the re of regio tht is isie the rioi r = + osθ outsie the irle r = 6 (swer: 8 ). 9 Volume of Soli 9. Volume B Cross Setios Perpeiulr To The X-Ais Let S e soli oue two prllel ples perpeiulr to the -is t =. If, for eh i the itervl[, ], the ross-setiol re of S = 9

10 Defiite Itegrl perpeiulr to the -is is A( ), the the volume of the soli, provie A is itegrle, is efie V = A A 9. Volume B Cross Setios Perpeiulr To The Y-Ais S e soli oue two prllel ples perpeiulr to the -is t = =. If, for eh i the itervl[, ], the ross-setiol re of S perpeiulr to the -is is A( ), the the volume of the soli, provie efie V = A A is itegrle, is Emple Derive the formul for the volume of right prmi whose ltitue is h whose se is squre with sies of legth. h. 9. Volumes of Solis Of Revolutio = f R.. Volumes Disks Perpeiulr To the -Ais Emple Fi the volume of the soli tht is otie whe the regio uer the urve = over the itervl [, ] is revolve out the -is.( s: 5 ) Emple Derive the formul for the volume of sphere of rius r. (s:.. Volumes Wshers Perpeiulr to the -Ais Suppose tht f g re oegtive otiuous futios suh tht g f for. Let R e the regio elose etwee the grphs of these futios lies = =. Whe this regio is revolve out the -is, it geertes soli whose volumes is efie V = f ( ) V = f g r )

11 Defiite Itegrl Emple Fi the volume of the soli geerte whe the regio etwee the grphs of f = + g = over the itervl [, ] is revolve out the -is.as: 69.. Volumes B Disks Perpeiulr To the -is.. Volumes B Wshers Perpeiulr To -is..e Cliril Shells Cetere o the -is Let R e the ple regio oue ove otiuous urve = f, elow the -is, o the left right respetivel the lies = =. The the volume of the soli geerte revolvig R out the -is is give V = f Emple 5 Fi the volume of the soli geerte whe the regio elose etwee =, =, = the -is revolve out the -is. Solutio Sie f =, =, =, the the volume of the soli is 5 V = = = = [ ] = Emple 6 Fi the volume of the soli geerte whe the regio R i the first qurt elose etwee = = is revolve out the -is. (Aswer: 6 ) Legth of Ple Curve, to = is efie If f is smooth futio o [ ], the the r legth L of the urve f L= + f = + g = = Similrl, for urve epresse i the form = where g is otiuous o [, ], the r legth L from = to = efie V = u ( ) V = u v L = + g ( ) = +

12 Defiite Itegrl Emple Fi the r legth of f = from (,) to (, ) Solutio f = f = + f = + = + The the r legth is efie L= + = + + l + + = 5+ l( + 5) If the urve is give i polr oorite sstem r = ρ ( θ), α θ β the the r legth of the urve is efie β β α α r L= ρ θ + ρ θ θ = r + θ θ Emple Fi the irumferee of the irle or rius. Solutio As polr equtio this irle is eote r =, θ The the r legth is L= θ = θ = θ = Emple Fi the legth of the rioi r = osθ If the urve is efie the prmetri equtio t, t, t [, ] legth of the urve is Emple Solutio () () L = t + t t Fi the irumferee of the irle of the rius r = =, the the Prmetri form, the irle is efie ( t) = ros t, ( t) = rsi twith t [, ] the L= r os t+ r si tt = rt = r Emple 5 Fi the r legth of the stroi t os t, t si t, = =.(s6). Are of Surfe of Revolutio Let f e smooth, oegtive futio o[, ]. The the surfe re S geerte revolvig the portio of the urve = f etwee = = out -is is

13 Defiite Itegrl S = f + f For urve epresse i the form = g( ) where g is otiuous o [, ] g( ) for, the surfe re S geerte revolvig the portio of the urve from = to = out the -is is give S = g( ) + g ( ) Emple Fi the surfe re geerte revolvig the urve =, out the -is. Solutio f = f =. Thus, S = + = = Emple Fi the surfe re geerte revolvig the urve out the -is. Solutio = = =. Thus, g ( ) g =, the S= + = + = ( + ) = ( 7 ) 6 9 Eerises Work out the followig itegrls 8. = l = + rt + si os = se θθ = 9 ( ) ( ) = =,

14 Defiite Itegrl 6. si = + 7. = l + 8. = rt e e + e 9. si = 6. os = 8 e. l + l =. e ( + l) = 9. = l z. z = 8 z = 5 6. = si = e 8. l e l = Fi the erivtive of the followig futios 9. F l.. = tt, As: l + t t, As: + t F = e t, As: e + e. F = os( t ) t, As: os os + Work out the followig itegrls. =. e = 5. =,( > ) +

15 Defiite Itegrl 6. = 7. l = 8. = = 9. = e l l. = +. = e + e. = l l Compute the improper itegrls (or prove their ivergee). 5. e >, 6. + l ( + ) e e rt

16 Defiite Itegrl lp 7. For p, is overget? (Hit: l p p for e ) 8. For wht vlues of k re the itegrls k l l 9. For wht vlues of k is the itegrl 5. Show tht = 5. Show tht 5. Show tht f f if e e = e = = ros 5. ( si ) = ( os ) si f f k overget?,( < k ) overget? ( ) f is eve f = if f is o. 5. The Lple Trsformtio of the futio f is efie the improper itegrl st F ( s) = L { f ( t) } = e f () t t. Show tht for ostt (with s > ). L t { e } = s. {} s L =. L {} t =. L { os t} = s s s + e. L { si t} = s Fi the first qurt re uer the urve = e (swer: ) 56. Let R e the regio i the first qurt uer = 9 to the right of =. Fi the volume geerte revolvig R out the -is. (swer: 8 ) 57. Derive formul V rius of se r. = r hfor the volume of right irulr oe of height h 58. Let R e the regio ove the urve = uer the lie = etwee = =. Fi the volume geerte revolvig R out ). -is, ). out -is. (swer: ). 6 7, ). 5 ) 59. Fi the re of the regio etwee = the lies = = 6. Fi the re of the regio oue the urve = si, = os = = (swer: ) 6

17 Defiite Itegrl 6. Fi the re of the regio oue prols (Aswer: 9) = 6 = Fi the re of the regio oue the prol = 8. (swer: 5 6 ) = + the lie 6. Fi the re of the regio oue the prols = =. (Aswer: ) 6. Fi the r legth of the urve 6. Fi the r legth of the urve 6. Fi the r legth of the urve = + from = to = (s: 8 6 ) + = from = to = 8 (s: 9) 6 = + from = to = (s: 7 ) 65. Fi the re isie the rioi r = + osθ outsie r = (s: + ) 66. Fi the re isie the irle r = siθ outsie the rioi r = osθ 67. Fi the volume geerte revolvig the ellipse Aswer: + = out -is. 7