Definite Integrals. Young Won Lim 6/25/15
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1 Definite 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 Differentils nd Derivtives (1 differentils rtio dy = f ' ( d dy = df d d derivtive dy d = f ' ( not rtio f ( + d f ( + dy for smll enough d = f ( + f '( d f ( + d = f ( + dy f ( + d f ( d = f ( + f '( d = f '( Definite Integrls 3
4 Differentils nd Derivtives (2 dy = f ' ( d dy = df d d dy = ḟ d dy = f '( d dy = df d d dy = 1dy = y dy = D f d Definite Integrls 4
5 Integrtion Constnt C plce constnt plce nother constnt differs by constnt plce only one constnt from the beginning dy = f '( d dy = f '( d + C dy = df d d dy = df d d + C y + C 1 = f ( + C 2 y = f ( + C y = f ( + C Definite Integrls 5
6 Applictions of Differentils (1 Substitution Rule f ( g( g'( d = f ( u du (I u = g( du = g ' (d du = dg d d (II f (g dg d d = f ( g dg Definite Integrls 6
7 Applictions of Differentils (2 Integrtion by prts f (g ' ( d = f (g( f ' (g( d u = f ( v = g( du = f ' ( d dv = g'( d du = df d d dv = dg d d f (g ' ( d = f (g( f ' (g( d u dv = u v v du Definite Integrls 7
8 Anti-derivtive? differentition derivtive of? f (? Anti-derivtive of f( Anti-differentition f ( Definite Integrls 8
9 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 Definite Integrls 9
10 Anti-derivtive Emples F 1 (= differentition All re Anti-derivtive of f( F 2 (= Anti-differentition f (= F 3 (= the most generl nti-derivtive of f( C indefinite Integrl of f( d Definite Integrls 10
11 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 Definite Integrls 11
12 Indefinite Integrls vi the Definite Integrl f (t dt nti-derivtive by the definite integrl of f( f (t dt d d f (t dt = f ( indefinite integrl of f( f ( d f ( d = F( + C f (t dt = F( F ( common reference point : rbitrry Definite Integrls 12
13 Definite Integrls vi the Definite Integrl f (t dt nti-derivtive by the definite integrl of f( f (t dt d d f (t dt = f ( indefinite integrl of f( f ( d 2 f (t dt = 1 f (t dt + 2 f (t dt common reference point : rbitrry [ F ( + c ] 1 = F( F ( [ F ( ] 1 = F ( F ( Anti-derivtive without constnt Definite Integrls 13
14 Indefinite Integrl Emples 0 f ( d = [ 1 3 3]0 = f (= 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 ( = indefinite integrl of f( d = C Definite Integrls 14
15 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( Definite Integrls 15
16 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 d Definite Integrls 16
17 Definite Integrls on [, ] view (I view (II G( = length 1 f '( = 1 g( = 1 re rbitrry reference point (, f( rbitrry reference point (, G( [ f ( ] 1 = f ( f ( [G(] 1 = G( G( Definite Integrls 17
18 A reference point : integrtion constnt C view (I Anti-derivtive without constnt view (II Anti-derivtive without constnt 1 d f ( = 1 d G( = f '( g( = [ f ( f (] 1 rbitrry reference point (, f( = [G( G(] 1 rbitrry reference point (, G( = [ f ( + C ] 1 = [G( + C ] 1 = [ f (] 1 = [G(] 1 = c f '(d c f ' (d = c g(d c g(d Definite Integrls 18
19 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( Definite Integrls 19
20 Definite Integrls on [, ] f ' ( G( re G( G( length view (I f ' ( d view (II g( d Definite Integrls 20
21 Definite Integrls on [, ] nd [, ] f ' ( G( re G( G( length G( re G( G( length c f '(d c f ' ( d c g(d c g(d Definite Integrls 21
22 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 Definite Integrls 22
23 Derivtive Function nd Indefinite Integrls f ' ( lim h 0 f ( + h f ( h f ( d f ' ( lim h 0 f ( + h f ( h 4 3 f ( d f ' ( 3 lim h 0 f ( 3 + h f ( 3 h 6 5 f ( d,, 3 [, ],[ 3, 4 ], [ 5, 6 ] f ' ( = lim h 0 f ( + h f ( h F ( + C = f ( d f ' (, f ' (, f '( 3 function of [ F( ], [ F( ] 3 4, [ F ( 6 ]5 function of Definite Integrls 23
24 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
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