Integration Danang Mursita

Transkripsi

1 Integrtion Dnng Mursit

2 The Indefinite Integrl The Definite Integrl Integrtion The Fundmentl Theorem of Clculus Appliction of Integrtion : Are between two curves

3 The Indefinite Integrl Definition : A function F( is clled n ntiderivtive of function f( if the derivtive of F( is f(x or F ( = f( Emples :. F( = + f( =. F( = 0 f( =. F( = + 5 f( = If F ( = f( then the functions of the form F( + C re ntiderivtive of f( The process for finding ntiderivtive is clled ntidifferentition or integrtion. Nottion : f(d F( C

4 Integrtion Formul...etc... C sin cos d cos sin d d.... r d d b. C d d d. r r C r d r r

5 Problems d 6 sec tn cos 4 5 sin cos cos d sin d d d sin d cos d d d d

6 The Properties of Indefinite Integrl cf(d c f( d f( g( d f(d Emples... sin : d d d g( d

7 Integrtion by Substitution Let y = (fog(. The derivtive of y with respect to is y = (fog ( = f (g( g ( [ The Chin rule] If u = g( then the formul of integrtion is f(uu'd du f(u d d f(udu F(u C F g( C

8 Problems d sin cos d sin cos d d cos d 5 4 d sec d sin d d cos sin d

9 Problem : The Definite Integrl How to find the re of region D bounded below by the -is, on the sides by the lines = nd = b, nd bove by curve y = f(, where f( is continuous on [,b] nd f( 0 for ll in (,b D = { (,y b, 0 y f( }

10 Steps : The Definite Integrl. Divide [,b] into n subintervls by choosing points,,,n such tht < < < < n < b. These points re sid to form prtition of [,b]. Let,,, n re the length of the prtition.. Choose the ny point,,,n in subintervl nd we hve Riemnn sum : f( + f( + + f(n n =. Increse n ( n in such wy the length of the prtition pproches zero (k 0 nd form the limit n n lim fk k lim fk k k0k n k n f k k k

11 Illustrtion : The Definite Integrl Y y = f( k b X X k- k

12 Movie : The Definite Integrl

13 Definition : The Definite Integrl If function f( is defined on [,b], then the Definite Integrl of f( from to b is defined to be (provided the limit eits b f(d lim n k0k f n k k lim fk n k k If the limit eits then the function f( is sid integrble on [,b] The vlues nd b re clled respectively lower nd upper limits of integrtion nd f( is clled the Integrnd.

14 The First Fundmentl Theorem If f( is continuous on [,b] nd F( is ntiderivtive of f( on [,b] then b f(d Emples : ( d ( cos d 0 b F( F(b F(

15 The Properties of Definite Integrl g(b g( b b c c b b b b b b f(udu (g'(d fog ( f(d f(d f(d ( f(d f(d ( f(d ( g(d q f(d p d qg( pf( ( 5 4 0

16 Problems d ( dt t t sin ( d ( d ( d ( dt t sin t cos ( tdt tnt sec ( d ( d ( d (

17 The Second Fundmentl Theorem Let f( is continuous on open intervl I nd I. If G( is defined by G( f(tdt Then G ( = f( t ech point in intervl I v( G( f(tdt u( G'( f v( v'( fu( u'(

18 Problems ( G( t dt G'( ndg"( cos t ( H( dth'( ndh"( 0 t 0 0 t ( F( dt 0 t 7 (find where F(ttins its minimumvlue (bfind open intervl which F( is only incresing ( 4F( sin tdt F'(ndF"(

19 Are of Region ( Let D is region tht bounded below by the -is, on the sides by the lines = nd = b, nd bove by curve y = f(, where f( is continuous on [,b] nd f( 0 for ll in (,b or D = { (,y b, 0 y f( } Then the re of D bis A f(d

20 Let D is region tht bounded below by the -is, on the sides by the lines = nd = b, nd bove by curve y = f(, where f( is continuous on [,b] nd f( 0 for ll in (,b or D = { (,y b, f( y 0 } Then the re of D is Are of Region ( A b f(d

21 Emple # Find the re of region tht bounded by the curve y = sin, -is, nd on the sides by the lines = 0 nd = π/

22 Let D is region tht bounded below by the line y = c, on the sides by the y-is nd curve = g(y, nd bove by the line y = d, where g(y is continuous on [c,d] nd g(y 0 for ll y in (c,d or D = { (,y 0 g(y, c y d } Then the re of D is Are of Region ( d A g(ydy c

23 Let D is region tht bounded below by the line y = c, on the sides by the y-is nd curve = g(y, nd bove by the line y = d, where g(y is continuous on [c,d] nd g(y 0 for ll y in (c,d or D = { (,y g(y 0, c y d } Then the re of D is Are of Region (4 A d c g(ydy

24 Emple # Find the re of region tht bounded by g(y = y y y, y-is nd on the sides by the lines y = - nd y = / 4

25 Are between two curves ( Let D is region tht bounded below by curve y = g(, on the sides by the lines = nd = b, nd bove by curve y = f(, where g( nd f( re continuous on [,b] or D = { (,y b, g( y f( } Then the re of D is b A f( g( d

26 Are between two curves ( Let D is region tht bounded below by the line y = c, on the sides by curve = g(y nd curve = f(y, nd bove by the line y = d, where g(y nd f(y re continuous on [c,d] nd f(y g(y for ll y in (c,d or D = { (,y g(y f(y, c y d } Then the re of D is d A f(y g(y c dy

27 . Find the re of the region enclosed by the curves y = nd y = 4 by integrting With respect to b With respect to y Emples. Sketch the region enclosed by the curves nd find its re by ny method y = + 4 nd + y = 6 b y =, y = - nd y = 8

28 Problems Sketch the region nd find its re by ny methods. y = 4, y = 0, = 0, =. = y 4y, = 0, y = 0, y = 4. = y, = y 4. y = -, y = 6, y = -, y = 4 5. y =, y =, = 0, =

29 Menghitung volume bend putr Metod Ckrm D (, y b, 0 y f (. Derh diputr terhdp sumbu f( D b Derh D Bend putr 9

30 Untuk menghitung volume bend putr gunkn pendektn Iris, hmpiri, jumlhkn dn mbil limitny. D f( Jik irisn berbentuk persegi pnjng dengn tinggi f( dn ls diputr terhdp sumbu kn diperoleh sutu ckrm lingkrn dengn tebl dn jri-jri f(. b sehingg V f ( f( V f ( d b 0

31 Contoh: Tentukn volume bend putr yng terjdi jik derh D yng dibtsi oleh y, sumbu, dn gris = diputr terhdp sumbu y Jik irisn diputr terhdp sumbu kn diperoleh ckrm dengn jri-jri dn tebl Sehingg V ( Volume bend putr V 0 4 d

32 b. Derh D (, y c y d, 0 g( y diputr terhdp sumbu y d c D =g(y d c Derh D Bend putr

33 Untuk menghitung volume bend putr gunkn pendektn Iris, hmpiri, jumlhkn dn mbil limitny. d y c D =g(y Jik irisn berbentuk persegi pnjng dengn tinggi g(y dn ls ydiputr terhdp sumbu y kn diperoleh sutu ckrm lingkrn dengn tebl y dn Jri-jri g(y. sehingg V g ( y y g(y y d V g ( y dy c

34 Contoh : Tentukn volume bend putr yng terjdi jik derh yng dibtsi oleh y gris y = 4, dn sumbu y diputr terhdp sumbu y 4 y Jik irisn dengn tinggi y dn tebl y diputr terhdp sumbu y kn diperoleh ckrm dengn jri-jri y dn tebl y y y y Sehingg V ( y y y y Volume bend putr y y V 4 0 ydy y

35 Metod Cincin. Derh D diputr terhdp sumbu (, y b, g( y h( h( D g( b Derh D Bend putr

36 6 Untuk menghitung volume bend putr gunkn pendektn Iris, hmpiri, jumlhkn dn mbil limitny. D h( g( Jik irisn berbentuk persegi pnjng dengn tinggi h(-g( dn ls diputr terhdp sumbu kn diperoleh sutu cincin dengn tebl dn jri jri lur h( dn jri-jri dlm g(. b sehingg h( V ( h ( g ( g( b V ( h ( g ( d

37 Contoh: Tentukn volume bend putr yng terjdi jik derh D yng dibtsi oleh y, sumbu, dn gris = diputr terhdp gris y=- Jik irisn diputr terhdp gris y= Akn diperoleh sutu cincin dengn Jri-jri dlm dn jri-jri lur y Sehingg D Volume bend putr : V 0 4 y=- 7 V (( 4 ( 4 ( d ( (

38 8 Metod Kulit Tbung Dikethui D (, y b, 0 y f ( Jik D diputr terhdp sumbu y diperoleh bend putr f( D b Derh D Volume bend putr? Bend putr

39 Untuk menghitung volume bend putr gunkn pendektn Iris, hmpiri, jumlhkn dn mbil limitny. D f( Jik irisn berbentuk persegi pnjng dengn tinggi f( dn ls sert berjrk dri sumbu y diputr terhdp sumbu y kn diperoleh sutu kulit tbung dengn tinggi f(, jri-jri, dn tebl b sehingg V f ( f( V f ( d 9 b

40 40 Contoh: Tentukn volume bend putr yng terjdi jik derh D yng dibtsi oleh y, sumbu, dn gris = diputr terhdp sumbu y y Jik irisn dengn tinggi,tebl dn berjrk dri sumbu y diputr terhdp sumbu y kn diperoleh kulit tbung dengn tinggi, tebl dn jri jri D Sehingg V Volume bend putr V 4 d 0 8 0

41 4 Cttn : -Metod ckrm/cincin Irisn dibut tegk lurus terhdp sumbu putr - Metod kulit tbung Irisn dibut sejjr dengn sumbu putr Jik derh dn sumbu putrny sm mk perhitungn dengn menggunkn metod ckrm/cincin dn metod kulit tbung kn menghsilkn hsil yng sm Contoh Tentukn bend putr yng terjdi jik derh D yng dibtsi Oleh prbol y,gris =, dn sumbu diputr terhdp. Gris y = 4 b. Gris =

42 . Sumbu putr y = 4 (i Metod cincin (4 y y=4 Jik irisn diputr terhdp gris y=4 kn diperoleh cincin dengn Jri-jri dlm = r d (4 4 Jri-jri lur = 4 r l D Sehingg V ((4 (4 4 (8 Volume bend putr V ( 8 d ( (

43 4 (ii Metod kulit tbung y=4 Jik irisn diputr terhdp gris y=4 kn diperoleh kulit tbung dengn y y 4 y y D y Jri-jri = r = Tinggi = h = Tebl = Sehingg y 4 V (4 y( y y y y Volume bend putr 4 (8 4 y y y y y V (8 4 y y y y dy / 5/ 4 0 (8y y 8 y y

44 44 b. Sumbu putr = (i Metod cincin = Jik irisn diputr terhdp gris = diperoleh cincin dengn Jri-jri dlm = r d y Jri-jri lur = r l y y y D y Sehingg V (( y ( y (8 6 y y y Volume bend putr 4 V (8 6 y 0 y dy 4 (8y 4y / 8 8 0

45 45 (ii Metod kulit tbung y D Volume bend putr - = Jik irisn diputr terhdp gris = diperoleh kulit tbung dengn Tinggi = h = Jri-jri = r = Tebl = Sehingg V - ( ( V ( d 4 ( (

46 Sol Ltihn Hitung volume bend putr dri derh yng terletk di kudrn pertm yng dibtsi oleh y =, gris y = 8 dn sumbu Y, bil diputr mengelilingi. Sumbu Y. Sumbu X. Gris = 4 4. Gris y = 8

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