INTEGRAL TERTENTU. sebagai P = max{x i x i-1 1 = 1, 2, 3,, n}. a = x 0 x 1 x 2 x n = b. Contoh: Pada interval [ 3, 3], suatu partisi P = { 3, 1 2 , 31

Transkripsi

1 INTEGRAL TERTENTU Defs: Prs P pd ervl [,] dlh suu suse erhgg P = {,,,, } dr [,] deg = < < < < = Jk P = {,,,, } prs pd [,] mk Norm P, duls P, ddefsk seg P = m{ - =,,,, } Cooh: = = Pd ervl [, ], suu prs P = {,,,,, }mempuy orm: P = m{ ( ), ( ), ( ),, } = m{,, 6,, } = Jk f fugs yg ddefsk pd [,], P = {,,,, } suu prs pd [,], w [ -, ], d Δ = -, mk [,] f ( w ) Δ dseu Jumlh Rem f pd y = f() w w w w 4 w w = = - 8

2 Cooh: Fugs f pd [, ] ddefsk deg f() = d P = {,,,,, } prs pd [, ] Dplh k-k: w =, w =, w =, w 4 =, w = 9 w = f(w ) = Δ = f(w )Δ = w = f(w ) = 4 Δ = f(w )Δ = 4 w = f(w ) = Δ = 6 f(w )Δ = 6 w 4 = f(w 4 ) = 4 Δ 4 = f(w 4 )Δ 4 = w = f(w ) = 9 Δ = f(w )Δ = 9 Jumlh Rem fugs f erseu pd ervl [, ] ersesu deg prs P d s dlh f ( w ) Δ = 9 Jk P = {,,,,,,, } prs pd [, ] d w =, w =, w =, w 4 =, w =, w 6 =, ser w 7 = 4 euk jumlh Rem fugs f pd [, ] ersesu deg prs P Defs: Jk f fugs yg erdefs pd [,] mk: lm f ( w ) Δ = L jk d hy jk uuk sep lg posf ε erdp lg posf δ sehgg uuk sep prs P = {,,,, } pd [,] deg P < δ, erlku f ( w ) Δ L < ε Jk f fugs yg erdefs pd [,] d lm f ( w ) Δ d, mk lm erseu dmk egrl ereu (Iegrl Rem) fugs f pd [,] Seljuy f dkk egrle pd [,] d egrly duls f ( ) Jd f ( ) = lm f ( w ) Δ 9

3 Jk f egrle pd [,] mk: f ( ) = f ( ) Jk = mk f ( ) = f ( ) = Cooh : Jk f() = +, euk ( + ) Peyeles: Bu prs pd [, ] deg megguk ervl g yg sm pjg Jd pjg sep ervl g dlh Δ = Dlm sep ervl g [ -, ] prs erseu dml w = Ak dcr l lm f ( w ) Δ = = + Δ = = + Δ = + ( ) = + Δ = + ( ) : = + Δ = + ( ) : = + Δ = + ( ) = Kre uuk sep =,,,, dplh w = mk w = + ( )= +, sehgg

4 f(w ) = w + = ( + ) + = + Jd jumlh Rem fugs f pd [, ] ersesu deg prs P erseu dlh f ( w ) Δ = + = + = + = + = ( ) + { ( + )} = + + Jk P mk, sehgg: lm f ( w ) Δ = lm = Jd ( + ) = 7 Cooh : Teuk Peyeles: Dlm hl f() = uuk sep [,] Aml semrg prs P = {,,,, } pd [,] d semrg k w [ -, ], =,,,,, mk f ( w ) Δ = Δ d Δ = -

5 = ( ) = ( ) + ( ) + ( ) + + ( - ) = = Jd lm f ( w ) Δ = lm ( ) = Deg demk = Teorem: Jk f egrle pd [,] d c (,) mk f ( ) = f ( ) + c c f ( ) Teorem (Teorem Fudmel Klkulus) Jk f egrle pd [,] d F suu uru dr f pd [,] (u F () = f() uuk sep [,]), mk : F() F() s duls [ F ( ) ] f ( ) = F() F() Buk: Aml semrg prs P = {,,,, } pd [,] Kre F () = f() uuk sep [,] mk F () = f() uuk sep [ -, ], =,,,, Berdsrk eorem l r-r mk erdp w [ -, ] sehgg F( ) F ( - ) = F (w ) ( - ) = f(w ) ( - ) =,,,, Dperoleh: f ( w ) Δ = f ( w )( ) = { F( ) F( )} = {F( ) F( )} + {F( ) F( )} + {F( ) F( )} + + {F( ) F( - )} = F( ) F( )

6 = F() F() lm f ( w ) Δ = lm{ F( ) F( )} = F() F() Jd f ( ) = F() F() Cooh: ( + ) = [ ] + Sol: Teuk egrl erku π s π Sf-sf: = { () ()} { ( ) ( + + )} = 7 kf ( ) = k f ( ) k kos { f ( ) + g( )} = f ( ) + g( ) Jk f() uuk sep [,] mk f ( ) 4 Jk f() g() uuk sep [,] mk f ( ) g ( )

7 INTEGRAL TAK WAJAR (Ipmroper Iegrl) f ( ) deg ervl [,] k ers u f dskou d [,] d mejd f d k u Defs: Iegrl Tk Wjr (mcm I) dmksudk seg: f ( ) deg f() erdefs uuk sep d egrle pd ervl ers [,] ddefsk deg f ( ) = lm f ( ) jk lm d d f ( ) deg f() erdefs uuk sep d egrle pd ervl ers [,] ddefsk deg ( = lm f ) f ( ) jk lm d Cooh: 4 e e 6 e Defs: Iegrl Tk Wjr (mcm II) dmksudk seg: f ( ) deg f() egrle pd [,] uk sep (,) ep dk egrle pd [,], ddefsk deg f ( ) = lm f ( ) jk lm d d 4

8 f ( ) deg f() egrle pd [,] uk sep (,) ep dk egrle pd [,], ddefsk deg f ( ) = lm f ( ) jk lm d + Cooh: = lm k wjr d s s = lm+ k wjr d s wh - Dlm egrl k wjr, jk l egrly d mk dkk egrl erseu koverge, d jk dk d, mk dkk dverge