Integrals. Young Won Lim 12/29/15

Transcription

1 Integrls

2 Copyright (c) Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version published by the Free Softwre Foundtion; with no Invrint Sections, no Front-Cover Tets, nd no Bck-Cover Tets. A copy of the license is included in the section entitled "GNU Free Documenttion License". Plese send corrections (or suggestions) to youngwlim@hotmil.com. This document ws produced by using OpenOffice nd Octve.

3 A f = slope = 4 slope = 3 slope = 2 slope = 1 f 1 = 1 f 2 = 2 f 3 = 3 f 4 = 4 Trigonometry 3

4 B f d = C Trigonometry 4

5 Anti-derivtive? differentition derivtive of? f ()? Anti-derivtive of f() Anti-differentition f () Integrls 5

6 Anti-derivtive nd Indefinite Integrl F '() = f () F() Anti-derivtive without constnt the most simple nti-derivtive F( ) + C the most generl nti-derivtive f ()d Indefinite Integrl : function of f ()d = F () + C Integrls 6

7 Anti-derivtive Emples F 1 ()= differentition All re Anti-derivtive of f() F 2 ()= Anti-differentition f ()= 2 F 3 ()= the most generl nti-derivtive of f() C indefinite Integrl of f() 2 d Integrls 7

8 Indefinite Integrls 1 d 1 d d dy + C y + C given vrible indefinite integrl d f d d c d f d d d f d d f ( ) f () f () f () f () + C given vrible indefinite integrl Integrls 8

9 Indefinite Integrls vi the Definite Integrl f (t ) dt definite integrl f (t ) dt nti-derivtive f () indefinite integrl f ()d nti-derivtive f () f () d = F () + C f (t) d t = F () F() common reference point : rbitrry Integrls 9

10 Definite Integrls vi the Definite Integrl f (t ) dt definite integrl f (t ) dt nti-derivtive f () indefinite integrl f ( )d nti-derivtive f () 2 f (t) dt = 1 f (t) dt + 2 f (t) dt common reference point : rbitrry [ F () + c ] 1 2 = F( 2 ) F ( ) [ F () ] 1 2 = F ( ) F ( 2 ) Anti-derivtive without constnt Integrls 10

11 Indefinite Integrl Emples 0 f () d = [ 1 3 3]0 = f ()= 2 f () d = [ 1 3 3] = f (t ) dt = [ 1 3 t3] = nti-derivtive by the definite integrl of f() t 2 dt = d d f (t ) dt = f () = 2 indefinite integrl of f() 2 d = C Integrls 11

12 Definite Integrls on [, ] 1 d f ' () d f '() = 1 view (I) 1 d g() d g() = 1 view (II) view (I) view (II) f ' () d [ f () ] = f ( ) f () g() d [G()] = G( ) G() Integrls 12

13 Definite Integrls on [, ] view (I) view (II) G( ) = 1 1 f '( ) = 1 g() = 1 1 d = f '() d 1 d = g() d = [ ] = dy = dy d = f ' ()d d G() = d Integrls 13

14 Definite Integrls over n intervl [, 2 ] view (I) view (II) length G( ) = 1 re f '( ) = 1 g() = rbitrry reference point (, f()) rbitrry reference point (, G()) 2 2 f ' () d = g() d = [ f () ] 1 2 = f ( 2 ) f ( ) [G()] 1 2 = G( 2 ) G( ) Integrls 14

15 A reference point : integrtion constnt C view (I) Anti-derivtive without constnt view (II) Anti-derivtive without constnt 2 1 d f () = 2 1 d G() = f '() g() = [ f () f ()] 1 rbitrry reference point (, f()) 2 = [G() G()] 1 rbitrry reference point (, G()) 2 = [ f () + C ] 1 2 = [G() + C ] 1 2 = [ f ()] 1 2 = [G()] 1 = c 2 f '()d c f ' ()d = c 2 g()d c g()d Integrls 15

16 Indefinite Integrls through Definite Integrls view (I) view (II) 1 d f ' () d 1 d g() d = f () f () = = f () + C = G() + C = G() G() = f () = + C G() = + C f () = G() = rbitrry reference point (, f()) rbitrry reference point (, G()) Integrls 16

17 Definite Integrls on [, 2 ] f ' () G() re G( 2 ) G( ) length 2 2 view (I) 2 2 f ' () d view (II) g() d Integrls 17

18 Definite Integrls on [, ] nd [, 2 ] f ' () G() re G( 2 ) G( ) length 2 2 G() re G( 2 ) G( ) length 2 2 c 2 f '()d c f ' ( )d c 2 g()d c g()d Integrls 18

19 Indefinite Integrls through Definite Integrls f ' () G( ) G() y = G() G() rbitrry reference point (, f()) rbitrry reference point (, G()) f ' () d view (I) g() d view (II) = f () f () = = G() G() = = f () + C = G() + C Integrls 19

20 Derivtive Function nd Indefinite Integrls f ' ( ) lim h 0 f ( + h) f ( ) h 2 f ( ) d f ' ( 2 ) lim h 0 f ( 2 + h) f ( 2 ) h 4 3 f ( ) d f ' ( 3 ) lim h 0 f ( 3 + h) f ( 3 ) h 6 5 f ( ) d, 2, 3 [, 2 ],[ 3, 4 ], [ 5, 6 ] f ' () = lim h 0 f ( + h) f () h F () + C = f ( ) d f ' ( ), f ' ( 2 ), f '( 3 ) function of [ F( ) ] 2, [ F( ) ] 3 4, [ F () 6 ]5 function of Integrls 20

21 Integrls 21

22 Differentition & Integrtion of sinusoidl functions d d f = cos leds f = sin d d g() = sin() leds g() = cos() f d = cos C lgs f = sin g() d = sin() + C lgs g() = cos() Integrls 22

23 Plotting Linel Elements single vrible function f () y two vrible function F (, y) tngent slope (, y) f '() F (, f ()) = f '( ) Integrls 23

24 Derivtive of sin() f () = sin() A slope leds d d f () = cos() Integrls 24

25 Plot of F(,y) = f'() (= cos()) (, y) = (, f ( )) = (,sin( )) y A2 f () = sin( ) slope (, y) y ' F (, y) = f ' ( ) slope m (, y) m = slope of tngent f ' () F (, sin()) = cos() Integrls 25

26 Plot of f'()=cos() from linel element plot A3 F (, y) = f ' ( ) slope f () = sin( ) f ' () = cos( ) f () = cos( ) Integrls 26

27 Derivtive of cos() f = cos B slope leds d d f = sin Integrls 27

28 Plot of F(,y) = f'() (= -sin()) (, y) = (, f ( )) = (,cos( )) y B2 f () = cos( ) slope (, y) y ' F (, y) = f ' ( ) slope m (, y) m = slope of tngent f ' () F (, cos()) = sin() Integrls 28

29 Plot of f'()=-sin() from linel element plot B3 cos( ) slope f () = cos( ) f ' () = sin( ) f () = sin() Integrls 29

30 Definite Integrls of sin() f = sin 0 / 2 sin t d t = 1 C1 0 sin (t) d t = [ cos(t)] re + 0 = cos()+1 π /2 sin(t ) d t re - 1 = [ cos(t)] π /2 = cos()+0 Integrls 30

31 Indefinite Integrls of sin() f = sin 0 / 2 sin t d t = 1 C2 f () d = cos() + C lgs Integrls 31

32 Definite Integrls of cos() f = cos 0 / 2 cos d = 1 D1 0 cos(t) d t = [sin(t)] = sin () 0 re - 0 π /2 cos(t) d t re + 1 = [sin(t)] π/ 2 = sin ()+1 Integrls 32

33 Indefinite Integrls of cos() f = cos D2 0 / 2 cos d = 1 lgs f () d = sin( ) + C Integrls 33

34 References [1] [2] M.L. Bos, Mthemticl Methods in the Physicl Sciences [3] E. Kreyszig, Advnced Engineering Mthemtics [4] D. G. Zill, W. S. Wright, Advnced Engineering Mthemtics