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M 12 30 3 00 PM

R f i oo 3 30 pm

Theorem Absolute property If feaca bthen Ifl ERca.br and

ISabfixsdx.es IfualdxRemark Ixtyle 1 1 191

III xile naI f

Isifixidxle fablfixildxproof Let Po fa xocx c cXw b be a

partition of Taib then on each Exia Xi

HI a P H I

II fautt iIt

Case 1 denote A Ixia Xi

suppose snap f z int f 70

so snap Ifl snap f

int Ift int fcase 2 appose ozszpfzina.tt

sosnap Ifl ingffin.at if I swap f

case 3 suppose SIP f z o z inaff

then Sfp Ifl maxf snap f inaff

guff o

mathI

inaf Ifl E min I snap f inafffft.Y.it eeoxstoEUglIfiLg Ift E Volf CflSince ft Ila b then as 8 I O

Urift Lrcf o

so by the squeeze theorem

Usc Ifl Ifl o as Tso

So Ifl e Ila bNotice that

Ifk l E fix E lfcxstv xe a.LI

by monotonicity of integrals

fabifcxsldxefabfcxsdxefablfcxsldx

elfabfcx.de ESabIfcxsldxA

Remark The converse is not true

e.gfix f

it to.sn

if KE Ion iQ

then Ifix I I C Iconbut f g Rca b

file fne kca.bl.tnb

An Safia dx Tff So if

th An Life fab fax d x

I f feefuxidxwhat is the condition for this to be true

Theorem Interchanging limit and integral

Let Effi be a sequence of integrablefunctions defined on Each Suppose that

f a f uniformly then f c Rca band

thy fabfax d fablesfax dx

fab fan dx

proof HW

Recall 9ft f tabs 77K

then we say f f uniformly ifHE o 7 NOTE s.t.vn

fncxi f ex I E for xo Eaib

we say f f point ifb Taib I 0 I 0 Sit

b n N N cx.es

fncxs fcx

fix1xEENxNkati

Example let Erny be an enumeration

of all the rational numbers in Coil

define Anti as it n l

fuk e Il it tri r mgo if xefo.dllri.rs e rn

fu C Reais t uz l

pointwise limit

fix if xo co.in Q

o i f XE Io 1 l IQ

f if Rlah

free fabfacade lez o o

7 Fundamental theorem of calculus

Relation between differentiation and

integration

Def n Antiderivative For f table 51kthe antiderivative of f is a

continuous function 112 sitF is differentiable on ca b and

data F let fix for all xeca b

Example i f tou Tk fix X

Text 31 3C

Remark If anti derivative exists theremust be infinitely many

Example f I i i Ph

if o X E I

f it.ieon o l x

on l l o C x I

Fix a 1 1 is not not differentiableat x D

f does not have an anti derivative

Hw Problem 17