MATH 201 Notes Part 4, Spring 2017

(pf ov20lb) oectare 24 (---catd) Boundary . ear jatlO, gicrjr0 D2a-3=9 AEH IT, 3 ( > - _ 2% ee | te: )9-34~2 49-97, hier) = Sta, hin) = 0.1 EAS = 3 D Boat 2 1 (420 D et 2 te 36AS3 3 1 3 ») Check value af alf the pods: We A(a,vehe-d 2 #0 =6-t& See 23 _~ ODa t ua ' 9 (-3) = (3,0) = 1 PHO = 2F eke aa a7 3(%4)-fByo) of -£ Hos MS J03)° 9- 7#0=l6 hl) 8.18 TKe are ACB js patemettised by 2> Scost, y=3sint OES DL feast, 3sint) =9-Foost-§sMt tO ‘ = (9-Feosk -bsiat OC ES g(t) = Pst — bcosk D FOO. beat>7g D b= Len" 25) Nose: Fiver tang =% Comp tet set | r cost -_! inal 42 |Domemuwhi ch tFclosed ead boundel bEKA EZ -3a-2y Ho P Extreme polues with coast ala s Civen Fad Pyke) 1 5109193 a “ireme Velues of f Subs ect to 9) (XK) Xr~Kn) =O re Hey ’ “J 92 (X, n> Te LAT | Le oy i SK Ny -- Xa) ‘ae ‘Arey, 2) - ty 422 COASET On LEYAP> et _55- (Ojov2a6) lecture 25 Method fdeg yaage multiples to find Extreme values of Feacttons oa Leve| OU VES oF Surfaces. -Find the eytreme values ofthe fuactron fcwy) =n YO vel the ellypse wey". 3 tiie . z r 2 - develcurve IS gC%Y) =A 43 -|=0 » The of Cry : Tf? thee js 4 b aint on the evel curve where fattans Qn extleme value, thentheghad eat off OF haf panty = VA my)= 75 fey) where is acoasteat Method i Co pu. v. + t 3¥ ¢ 5 y) ( “C V+H, ard 6 lve h Ue a) fo n [ 4) 10) “ A , lution 3 7 2 Solution: flay > yPeap hee Vay = A Pilg) 15 (44) 20 u oye ae, a=hy and Ay j20 Note: TP\-0. ap. ae Serene Hieeatai Ae / ak 4 would he Z2elo which isirp osst ble MG tothe thitd eguation .” \ £0, D dyed aysdgr so dt, ay thus ogy" graay 2 rr [seep #2) = == [=O » JHLl and a=tr [4p oir s) A(2,1), B02, 1), ¢(2,-), 0(-2,-D check for all the poets: | ies (29 AOI 22 Jed AB) -2 J doug), --2 Proof of Teo ten (dea) Ne gf M sts ch fut Come c ( rug) =o) NomMal (S the Same ©) ? Expect fry) =C ond 90m y)=e to tough at Lpsint [04 TANIA] (ANOV20(4) decture 26 Mul tipe Zrtegals “A pplicadtioas 7 | ) Finding nfeas volumes Momesty center of Sffeug da 5 mesPycanity, nomen! of Adertia, higher . moments (osed in ol ett ics) Il [ for 2V first wXge bace te one variable. n / ay eee ? poe mn 2 4 a b [feos = Limit of Ri emeng Sums a 1 How do we calculate t 7 Will Study ‘ Itc BX 1) itf(a)ar rt... Hn) d% Fukdapete) ttecian of olouter Pending prem tives CNeVten) [utdn 4 git nd-/ 2 First Lites a primitive Fotf thet nt iY Fla) = fea) _t i b jMaki [fore = Foo)’ = Fl) fey) a a Lz of i fy sda= at[?_ 2 1 _ “ |* fi a] Fe ~ ert w ° det fre Qa furction ‘of Qvertebles J CMM A)dr a ree e* C1) defined on the domain Doan xcyplane, . Qi emMGnn Sam} > fOr; 4) budy. LP Li mit of IUCh sums exist is called pe SOULE | A1e4r al of f ever D L Denoted aaa) aya D Lb JAN20t6] Eater: OA (y=0 , Osx) (ntl) | Exit! ©B (4=4,° 08 rs) How to Compute docble iteg als 7? pe Disgavec ) ‘ Suppose (§ @ ang Le Eater: 0B (124, osy<p 0 - (Ac) | AC XSb | cs yid} Exif : AB (asl eS 4) and 1S delired on D SA approach ’ l “ if [7 © Pin ff §%5 e . b or Ew approeh pon [Beat te MP9 0 c J cho ase (-f/ ApplOach because jf iS Caller ’ a cin Pact Ew pp! Sack Bano be dont) Examgle : = aaa | D= ftp] Srl, 08 6 2} ; ced a etash} ic roth ft1,4) = 1% 45 2 1 = [~cos] 3 then SfPer) dA [Ye yrdady ° Co NSi der S [fersid 4 whey & 0 ¥.°3 D 2 | f wii k =f 84) dy = [[dojdy 0 = flagyl Pers, 520} J J ¥ J ie ‘ 34 Set op cts Hmripef integratmy b = 31° 2% oritcer ave (i) sketch D: = 2 vedone au! 6 (0;2) ° [Cl oe Sduide b CIN IG ey 44) | B20) t ge. 6 2,0 chia fheorep, .” Nee eel etvalute wh I rom r 2a Eval LP Sa% dy aes Ente: AB Y=, 2S oY Ls =f 2<ae2) the triangle ofvertites O[0,0) mF KB or 7 . fat Atle) B (lI) ~ avpsy = 10 4) d du «Sketch of D J Jp gine itis Not @ V7 ) fei gagle = dace be tle fesi@n v.¢ J OE north appl02ch of east vest ager ach Oe) (4 0) E-wWappre ack E-wW -W approeck . Arter At (X=-f4-p) oS 982) iar d Exit: ¢B Ae ice OS 92) fe aa cm =0.°(3,- 2) Ei Ater from 2 4i flere (ia cive [tery “ff frag)d ay Jor 2 ky cl ye Poe d(egte) Oo “Thy Crust bt writfern (n 2 te ° lived meen e 2 D4-2-3-y =O se Note: A= lL+fi3+4 S-M approrck hie dy dy 4 Ester ys N3+4 E-W atone “Ea ae Lhd 6 IS -| [porsiinry ; Leg Beg 2 bye y Examole : Eydluate the Micgla| Und At Where fairy = ot 7£ tetegialf F 1 A-Y -l ey 3 (09 100 Qovierd ye =g and uF gm T1344} ~1< -) 14 fiFRy 5 2 x Je 3 - find INE receCtion of 4 =% andy wtrza3 J : "Em 7 teh th (rtegfa te ACh), Gy Problem: Stetch the regiva of /shegra fe For S-y Rpproreh enter Crom bleu gia ey dx j= a%1-3 , 1A %ts3 and Wtf: yet, -[62 <3 ° 0 Answer.’ Eater yo, e677 | Exit yoy ) OS xé/ e-L-ti% I Gfereyejes (4 piss © t-3 : #Mtegia-tion) 2 rfryeF] dt. ee 4-4-3 Problem: Taterchange the order S-V applowh Enter a Jr-lray, ra Asa Ex} 5 y= the, TA 22S oe Iie * a 2 42 = 2 fara de = 4 [RI Ix “a KRostacecl Oo To evaluate HUT (A469 el) cd an tly fochange chaty PVen Purction UA der the spuare (vot to be @ perPect Sguere : ws Osta =| D a0" = a arate Put 22 2C0O \, dasa sinQdeO D Gi. 94 4 tla 9? dx ak d 2 ws O(a OPO- (3) qT _% x yaedy SJ OD fat [eis . To 2 = 2a (B~ s1n20 J} - War 2 0 Hy Ov201b) [oF Tan 212] oectare 2Y vy a has ai nde coordi fe {This is called {Merchaage of , Limithg pes ahoas ie | 0m = pm f! cays from orga 7) | LEE ate. , feplicaticas » fleas oO enlarge Riven aegion Din xgylene Ares of eee [fans foe ns 2 Frample' ae fhe arfa of di tre (0,0 8 radu 0. aia and A Sketch rey rag? (boundary o §£ irec= aco - OAK Ln - Ave of Fécjur = KA™ Descrloe€ D 2 (5) . . Eater: rf -E <6 ., afta a (Jae -L 720 rae (> tees ar = CA-Ae@ +4 (Ar) ae [Carb A Riemann Cum: dh dido = yee uf Lit 0) - Looe aA ¢ terms w/t (24) ag” in dhe st Whqe p) rdedé pao Lye eHow to Set wp Umit of integration “h g, ‘ —~ : = 2 ((2csoter6d6 Ente ral, OSO¢ -%, Ext rh / eaek. = treat. ireer2P 46 A bs & £ aes “Z, Lecter < Lpasinoed + 020 | ? i} >= Barges - = 3 [C2 *# +0) -(-2-y- 0) J - 2 +7 ed 2,0) * find afta poured by 4he Benn sca Fe r= AR 2cpes 26 & sine) rdtd& bch (eG Fife sin &) Mek ~ 4 = (pat 20) Koyert Find area of region iaside cardiod ra ltlve ancoubide ewcle C=] e ; 4 fymnrell § 5 Oe ae ses aecot | t pucadraat. HeScribve D: Enter: £ =O, 0s Os Y 2 ae (= 1a bead ose ray fesse we " 2afos28 dO hrer= [fda D What abu? .3 -9 4 +2 f 4e?cos.6 de oda or mo /e 7] % Chrepall © f° _ a = B Q2 ty, 2OdO s) v| | eé of ? Lavcolutionr] 5 : eae: |6MOV20l6) = 2 6(a2e7 F ( olecture 24 aCe’ & 2 i = tae » Find the volume of a sphelet of radius a, HiStotecal- Filieate . We ma aseune Hat ceatre of phere cs bo, atthe eigin O60 ity Spur tion: [ews 2 t4?€ B= ar a po , of a) Sdution: Lef Te fe" As and evaluate 7Z ‘Trl aa L in) fer? * CE? u a rd) bg Tre [pt ep ny [evr - eed) de * bs oo dea eo aa ys volume ef the opper 7 ry? emisphere “Hf 3 = {fer dy da Need to sketch Din b [a> n2y3 dA 20 the x-y plane. jin Region of ate glad lon Disadise watie Dr J ia polar? git quadrant =) Eater’ r=0, 0s0S3, ca) o) P Exit : rao ,0SOSY, amawang tT? PO x£ \ i p star 12: (7AM = > ; ; C rdrdé = [te jae Enter 0, OS O<2y O 2 xt Fre) oS Os 2¢ 0 9) ec ? Volumes oa -4 _ tse * I [euc rare “Who -(-4) Jade = 3 = ay f a fap?! ? TT 0 Att Waza => = -s G 400 Fe eoae \ us fae rdps ae - fe*daz a ? Je mW odo 2 ] 4 7 \ gp fe da - | This is the noma| 7 fo,..0c8.. dicthrvbudien oon 2T a2 2% a Exif: surface Sy (2=93Fcr, y)) 3 - a f udu = 12 “| I = (ny) iS in region D plojection of fac 3 o the solid Roan the “y plane 40,3) 7 24 le a)WV |fpen2ye2Md Og uy a od Vola me o- = ° 2a%(2n) ate? FR Liev Ai chimed Sow : a ae d// where KR tue cold hoon de) (chi id ok adore Ry Sf, ee ia and be(0Y QE AA EEE Epes oly err - ¢ Tple jategéals: 7 / 3 variables fate! tea ty? (nist Ww=F( 9,2) Exch: 2=9-n%4? (919) inD ; \ ; _ Di 2= Rig avolld regidata 3 «pee Ponte g-aby > Sh ( ee PIE yi Dd V as Y k it > Dis the dice in 79 rN To evaluafer fale iA tegra | bounded by the cifele ahrey? ay t ap dgr Os ketek Dif dll weasels’ {fay [I(r tedan a a nspag, ; - 2 faa? F-A yr ff [rede [6-272 Mh =e ary 9) ‘Lavelface lige 47 ; (2) seca” ra be p eicrib Cin p lar: Enter: Sarface £ ONeay" =f feo-2 race re. \ fo (gwouaels) + Re dues nee called Pil st moments (ay yh, 2') Moméats c Y a omenf= Me \@) = + One f ond | ae Sel US y 0) I @® f *Ceeka poi (x9) so thed c «|| the Mass of D tJ conce At thes afb Psumup if D Lane £4gi?a 9 wlth Lonely pes ctiog fing at (ay) J [tough thea: Moment DO Wr, } yo axis [fader 4? 7 _ Sig tery and ears fea yds 0 J pQ Si wiley pore nd wilt H- exit /['yha pon Lease ‘det R be a solid iq 3 “space D Tr 3 } ls ' d the com ofa colid R ac Esse Ak) aR ty daca g Mas == [peer yay ee Ege) : K = fffasrpouv : Moments in 3d re abe Wy [adc rey, aav= Mya WAS lv R R ie flay, 2)b/ = ire: Note: whe veal hr ve of gi evity(C0¢) (x4) = R Example Anh Coe li hourde] by the J [feds zal >May here Vey tary qz and the 1-9 plane ithe Upper teat space. R _ AA