this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]

Transcription

1 Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow s tegrto. Defto of Atervtve: A fucto F s tervtve of f o tervl I f F ( x) for ll x I. Represetto of Atervtves: If F s tervtve of f o tervl I, the G s tervtve of f o the tervl I f oly f G s of the form G(x) F(x) +, for ll x I where s costt. G(x) F(x) + s clle fmly of tervtves or geerl tervtve. s clle the costt of tegrto G s lso kow s the soluto to the fferetl equto A fferetl equto x y s equto tht volves x, y, ervtves of y. Ex: F the geerl soluto of the fferetl equto y Notto for Atervtves The process of fg tervtves s clle tfferetto or efte tegrto s eote y tegrl sg: y So from y usg tegrto o oth ses of the equto ths s the efte tegrl y F( x) + Sce tegrto s the reverse of fferetto we c check the F ( x ) + f ( x ) prevous y [ ] If you kow your ervtve rules the lerg your tegrto rules shoul e very esy! Just work ckwrs.

2 Bsc Itegrto Rules: Dfferetto Formul Itegrto Formul [ ] 0 0 [ ] k k kx+ [ ( )] kf ( x ) k k [ f ( x ) ± g ( x )] f ( x ) ± g ( x ) [ ± g( x)] ± g( x) + [ x ] x x x + + s x cos x cos x s x t x sec x sec x sec xt x cot x csc x csc x csc xcot x cos s Ex:. 3x. 3 c. x x f. ( x+ ) g. 4 3 x 5 x + x ) h. sec x s x+ x cos x+ x t x+ sec xt x sec x+ csc x cot x+ csc xcot x csc x+. s x e. x+ s x. x cos x

3 Are: Ojectve: Use sgm otto to wrte evlute sum. Uerst the cocept of re. Approxmte the re of ple rego. F the re of ple rego usg lmts. Sgm Notto: The sum of terms,, 3,, s wrtte s where s the ex of summto, s the th term of the sum, the upper lower ous of summto re. Ex:. 6. Propertes of Summtos:. k k Summto Formuls:. c c 3. Ex: Evlute ( k + ) c. x k ( + )( + ) 6 +. ( ± ) ±. 4. ( + ) 3 ( + ) 4 for 0, 00, 000, 0000 Are of Ple Rego Use fve rectgles to f two pproxmtos of the re of the rego lyg etwee the grph of x the x-xs etwee x 0 x. Rectgles outse the curve re clle crcumscre rectgles the sum of the res s clle the upper sum.

4 Rectgles se the curve re clle scre rectgles the sum of the res s clle the lower sum. For y rego uer curve f oue y the x xs etwee x x. () The left e of the rectgle touches the curve () The rght e of the rectgle touches the curve where x, s the umer of sutervls f (m ) f( + ( ) x) f(m ) f ( + ( ) x) f( m) x f( M ) x f the fucto cresg or ecresg wll chge whether () or () re upper or lower sums f (m ) s upper sum f f s ecresg lower f f s cresg f(m ) s lower sum f f s ecresg upper f f s cresg Lmts of the Lower Upper Sums: Let f e cotuous oegtve o the tervl [,]. The lmts s of oth the lower upper sums exsts re equl to ech other. Defto of the Are of Rego the Ple: Let f e cotuous oegtve o the tervl [,]. The re of the rego oue y the grph of f, the x-xs, the vertcl les x x s Are lm f( c) x, x < c < x let c + x Ex: F the re of the rego oue y the grph x xs, the vertcl les x 0 x. 3 x, the

5 Rem Sums Defte Itegrls Ojectve: Uerst the efto of Rem sum. Evlute efte tegrl usg lmts. Evlute efte tegrl usg propertes of efte tegrls. Defto of Rem Sum: Let f e efe o the close tervl [,], let e prtto of [,] gve y x 0 < x < x < < x < x where x s the wth of the th sutervl. If c s y pot the th sutervl, the the sum f( c) x, x < c < x s clle the Rem Sum of f for the prtto Defto of Defte Itegrl: If f s efe o the close tervl [,] the lmt lm f( c) x exsts, the f s tegrle o [,] the lmt s eote y lm f( c) x The lmt s clle the efte tegrl of f from to. The umer s the lower lmt of tegrto the umer the upper lmt of tegrto. Notce the smlrtes etwee the efte tegrl the efte tegrl. Eve though they re smlr there s mjor fferece the efte tegrl results umer the efte tegrl results fmly of fuctos. Ex: Evlute the efte tegrl x rememer x x c + ( x) otuty Imples Itegrlty: If fucto f s cotuous o the close tervl [,], the f s tegrle o [,].

6 The Defte Itegrl s the Are of Rego: If f s cotuous oegtve o the close tervl [,], the the re of the rego oue y the grph of f, the x-xs, the vertcl les x x s gve y Are Ex: Sketch the rego correspog to the efte tegrl: 3 Deftos of Two Specl Itegrls: z. If f f efe t x, the we efe 0. If f s tegrle o [,], the we efe z z Atve Itervl Property: If f s tegrle o the three close tervls [,c],[c,], [,] the, z z +z c c Propertes of Defte Itegrls: If f g re tegrle o [,] k s costt, the the fuctos of kf f ± g re tegrle o [,], z. k kz. [ g( x)] g( x) z ± z ±z The Fumetl Theorem of lculus Ojectve: Evlute efte tegrl usg the Fumetl Theorem of lculus. Uerst use the Me Vlue Theorem for Itegrls. F the verge vlue of fucto over close tervl. Uerst use the Seco Fumetl Theorem of lculus. We hve looke t two mjor rches of clculus: fferetl clc (tget le prolem) tegrl clc. (re prolem). Eve though the two seem urelte there s coecto clle the Fumetl Theorem of lculus. 4 The Fumetl Theorem of lculus If fucto f s cotuous o the close tervl [,] F s tervtve of f o the tervl [,], the f ( x ) F ( ) F ( )

7 Usg the Fumetl Theorem of lculus. Prove you c f tervtve of f, you ow hve wy to evlute efte tegrl wthout hvg to use the lmt of sum.. Whe pplyg the fumetl Theorem of lculus, the followg otto s coveet. ] F( x) F( ) F( ) 3. It s ot ecessry to clue costt of tegrto the tervtve ecuse f ( x ) F ( x ) + Ex: Evlute ech efte tegrl [ ] [ F( ) + ] [ F( ) + ] F( ) F( ). ( x 3) 4. x c. sec x. x 0 0 Ex: F the re of the rego oue y the grph of y x 3x+, the x-xs, the vertcl les x 0 x. The Seco Fumetl Theorem of lculus: If f s cotuous o ope tervl I cotg, the, for every x the tervl, x f() t t Ex: Evlute x cos tt 0 Itegrto y Susttuto Ojectve: Use ptter recogto to f efte tegrl. Use chge of vrles to f efte tegrl. Use the Geerl Power Rule for Itegrto to f efte tegrl. Use chge of vrles to evlute efte tegrl. Evlute efte tegrl volvg eve or o fucto Ptter Recogto: We wll look t tegrtg composto fuctos two wys ptter recogto chge of vrles. Rememer the h Rule: [ ] π/4 f( g( x)) f ( g( x)) g ( x)

8 At-fferetto of omposte Fucto: Let g e fucto whose rge s tervl I, let f e fucto tht s cotuous o I, If g s fferetle o ts om F s tervtve of f o I, the f( g( x)) g ( x) F( g( x)) + If u g( x) the u g ( x) f( u) u F( u) + Recogze the ptters tht the followg ft f( g( x)) g ( x) Ex: x( x + ). 3 x x + c. ( x + ) x e. 5cos 5x f. sec (t + 3) x x x( x + ) Mkg hge of Vrles. hoose susttuto u g(x). Usully, t s est to choose the er prt of composte fucto, such s qutty rse to power.. ompute u g ( x) 3. Rewrte the tegrl o terms of the vrle u. 4. F the resultg tegrl terms of u. 5. Replce u y g(x) to ot tervtve terms of x. 6. heck your swer y fferettg. Ex:. x. x x c. s 3xcos3x Geerl Power Rule for Itegrto: If g s fferetle fucto of x, the + [ ( )] g x [ ] Equvletly, f u g(x), the g( x) g ( x) +, + + u u u +, + Ex:. 4 3(3x ). 4x e. ( x ). ( x + )( x + x ) c. cos xs x 3 3 x x

9 hge of Vrles for Defte Itegrls If the fucto u g(x) hs cotuous ervtve o the close tervl [,] f s cotuous o the rge of g, the g( ) f( g( x)) g ( x) f( u) u g( ) Ex:. 3 x( x + ). 0 5 x x