2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture...

§$,2- The- Defil'1 iu Intej ral Reo;di . I r ~ r.:f) Sh - ,yt ~ ~ tt j{tJe. -JrlL =- tt f(~OtJ.:;( ,f(?C) ~ tl ,tt).Y1 c-n'tm- det:~ on en. U1ferva / [C?J b] (() D tvr d e. (. tt I h] ut-to n. ett.{Q ( SU.hU1 i:e'wals M.= b-;:-:efd fh ue s:"ub vvtf£Y1/als) D;Vj'Sl'VfL POlttts a.~ 5(0 <: :it < 9L 2 <-- < 9(1< < r r - <~ -= b urh eYe.- ~ = a -l- Ie--.4?L (2..) c.,hO{)5~ .R~ betweot ~_, evtA i k anA tJ-tf'Y'd)(fJ4l4fe.- ~ o.v-ea. 1 ~H!sfri p hy

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Transcript of 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture...

Page 1: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

§$,2- The- Defil'1 iu Intej ralReo;di . I r ~ r.:f) Sh

- ,yt ~ ~ tt j{tJe. -JrlL =- tt f(~OtJ.:;(

,f(?C) ~ tl ,tt).Y1 c-n'tm- det:~ on en.U1ferva / [C?J b](() Dtvr d e. (.tt I h] ut-to n. ett.{Q ( SU.hU1 i:e'wals

M.= b-;:-:efd fh ue s:"ub vvtf£Y1/als)

D;Vj'Sl'VfL POlttts

a.~ 5(0 <: :it < 9L2 <-- <9(1< < r r - <~ -= burh eYe.- ~ = a -l- Ie--.4?L

(2..) c.,hO{)5~ .R~ betweot ~_, evtA ik

anA tJ-tf'Y'd)(fJ4l4fe.- ~ o.v-ea.1~H!sfrip

Page 2: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

~ ~ t-(1:)LJ7( c

k-=--I

11I, i; ~ .f-(?f..tJ~J...~ --7 (>0 Ie -=-/

e)(j'S'-ts! f'ot'" aYJi ch"iLeo i ;;{;-:thu~ t-i-. -I-hi'S (aYe

II ~ RV\ :: l,~ LL1 ;=!- 1, -. NYI //Yl-?{)O n."'::'(JO I'l~VO //

--rh~5 /,hi if (S cal leo!.. ~e dcl'Vl'd:e vrtte~ralbff-lWn.. a.. To b\~{X)Ji 2: ~~<AIA1IYtt\-f;~ 5; __ jJ~ <1.: lower {'m-i~ .J ~ (.,vj,feqrqj

Cl tA h! tApper IIIMf "6 ()~f~rM- ~j. _ » d:;(

th.....,,~

Page 3: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

G-.eeJ11U:+n'G m e.a. Yl iy ~ of -fhe. de-11 n i-te IX f.e.:roJ

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(2.-) ,f(i) ~ 0 ern (~I bJn

} \ l) -TheN\ i: L..f.rj:)'i.:lJ

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J(:::( to lu Wlt1> /

lab;} i1rni1; rk " (' )J~)d;;(J~~ t/[;(:)&k - AreA ~1)

(3) G-,e.rterct IWe-, ct.fc-;() C&lhl be ?O or ~ 0 .- / -,'

Page 4: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

Page 5: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

~ b

q b h b\t.f(7{)-~(/ojbl=-1-:KX)4i- i :}(7<)d?(Jo. o; o:

Page 6: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

Page 7: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

S~J~z; ~r~(~)o 3

~S5C' kl-=-I~~ I f(;,(:)A;(6 1t t--l

U/.£- 1: =' ~k: J n'Jhf ~ 1lJt'le-l)I(~JY\

==-{J ~ 'Lft(~},1J(~-) ()O

Ie-=-rL- -y-- J

Ri'ema.nt1 sum

Page 8: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

Yl ®~ k -c I+ 2.-+ 3+-- -f 1'L -::: !Len +,) IIe"'-I .2. <le-3r.ee '2...

t k2..-=. t-+ 2~ 3~ -- + 1l\:. IJ!JlLfJ(2J1-fI) .kJree3It -::> I+;/= J3+-i-f 3'+ _ + ?1!=- ) 1'fL{1l-f1 ) 2-L-.J l.2 ..1JreefIe~I ',

,±Jt -::l.t-r 2.€+3~ -- -f- n,( '" ( kg~e .U! POJ>mon1t:.t)led (,\' 1'L .J

13ack -to fu lHt~'JYCtl.

i -t-MIe).tYL =- 3~ i k! ~ ,3~ '(rt-fQ~!-{ ®+i)2.Jt-~ I I/tt k~l n"'i: 2... 2~ -T 111.. -

~ ~(~)2-~ ~ (1+ k)2t:t3J~::::II~ if(~Jc-)&l =--II~ ~Ut 1~pi

a 41~(.JQ L)e"""~ tI.~OO 'f- /h-" -fRieJt1.?tYl'll j'u,VI1

Page 9: 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture Notes Math 11B/Section 5.2.pdf · §$,2-The-Defil'1 iu IntejralReo;di . I r ~ r.:f)

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Transcript of 2- The- Intej ral Reo;di . I tt Sh r ~ - ,yt ~ ~ j{tJe ...tamanoi.math.ucsc.edu/tamanoi/Lecture...