S 16-4 - Math 200 Spring 2010 Worksheet#26 Name K92 Section...

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Unformatted text preview: Math 200, Spring 2010 Worksheet #26 Name K92 Section 16-4 Parametrized Surfaces and Surface Integrals 1. Identify the surface with parametrization CD(u,v) = (u sin 2v,u2,u cos 2v). Pow ‘ ' . .. ‘ : QMN’HC Qtlflflfl‘mhf are. x._ {ls-‘33”) yth) a: “CaJJV 1 For an} earn- on 44w Svri’ace ) 3941‘..- u‘mhuq. “lags'llu : u :7 Space +here are m Poli’flci-ion-I on ‘Hse {)qmméli’rr +1.9 SUrFQt: L - - - ‘1: XE”; IS «a Cll‘t'biax par-abated whose Orin; is He y’awis- 2. Use Winplot to graph the parametric surfaces and indicate on the graph which grid curves have u constant and which have v constant. (a) CD(u,v) = (usin2v,u2,ucos2v) , 0 S u S 2, 0 5v 5 2:: hoidtiu} M. Conx’mni ) § (m, V): (”a Siniv) ”01.) {JD Casi”) NHCPA are Grimm 1;. H»? yicowhwt Planes. H 3"“?qu u}; u COSQVg) which are QY'Glo \ - (b) @(u,v)=(ucosv,usinv,v),—5$u$5,05f527r alas 59mm°f"“ aLod— ”‘9 3‘4‘15 . hairless U. Cas9i'mr} @(QoJU) : {Lie (05V! HQS—InUJU) win-cl» giufl Q. Lek? inciting U (.Onii'an't’ Ely.) Va) 7. (u (051)., U. (in Va, U0) Wind‘- 8N3! Sl‘mtéri" uneS. huidllfi U ConS'temi‘, 50’5”") : ( 3. Find a parametric representation for the surface: (a) the part of the sphere x2 + y2 + 22 =16 that lies between the planes z = —2 and z = 2 . A famine-Hi“ refreSGni'oi-u'aa UFO. sphere is? Padiur 1-} I §(@’@) 1' (‘1’ S'MCPMS 6/ ‘1‘ SIM? Slime) 9(016’) The store 'mEv-seai-s .piane i=2 W'H’m 2- :: “from—1’2 :) (“993% 3) (132% 37 ”mm-err}, intersect-Can. w'th 2-=-'2 Correspondr‘h: 43: Ni? ‘-‘~ ”9M, it? SVrf-oca. is described by EMW)={‘f$in49(056‘;’151h¢7$:}r9 Word? , (b) the part ofthe circular paraboloid x+y2+z2 =4that lies in front ofthe plane x=0. of 95511-3 V35!” 2% Taking 7 and i 0.; Parameters) E7012): ('1'~77'—%;'))%) Wit??? y2+ilmél1~ . 515a? x 7 o z: - ‘ 1 "- Iii-lac) r?- 7 4 (.056 and 2: — .0 (me , 933—4, :: «010576 +£23.38 24“ That) §( 1. . 4,9») 7- (“ha , Mose, aslna),o:e$2n~) o M 61 (c) the part of the plane z=x+3that lies inside the cylinder x2+y2=l. 05““) _ 4 Mg 6 as funeral-em, >8}, 1 = 4160929 +‘OLSIhz-5 :5! Lk 44‘ . J 109;; 415i :) 056$. "‘1 e whoa“ ‘5 9“" 5) N456) TfACOSBJASin6)0Co£6—LS)) 05:15.!) 05-65211“ 4. Find an equation of the tangent plane to the parametric surface at the specific point: 2 2 _ = = 3. x: u, y: v, 2: av, u 1,121 §(u)u):(q)u:uuu) / EH”)? (1,1,!) g “-_.1 11(1):)» T“(2MJ[ U 112-) (2Mo|‘>x(o 2,17 “73“: (01mg :C5$-(J TR”) Gt”‘93"“44” °m"*““3'*f|m¢cmn 33 054 ‘ "9 ’3 “‘7 “)(X 0 1(3—1) t-q-[}-1\—t> <"‘> “JI-Jy-Hfi: o 45‘) ;<+-y~ 29:0 5. Consider the parametrization (13(0, z)— (Rcosfl, Rsin6,z z), O<t9<27r, —oo<z<ooforacylinder of radius R with equation x2 + y2 :1?2 Compute T6,Tz, 11012), and "n“. z—grid curve To ifizagmcose RSmG Ii“)- { Rsme Rmsfi 0) T}: %§3L(RCOJS/R31neA9-): ( 0, 0/ ' > ‘ a: _ i. j k . -_._‘ N61“) Text: " 4257.6 9st o _‘-‘ { RCUSQRMQO _ . .4 _ x 2 16 O O I in: s) ishofimu. 6_gndcme . 3 '0“ ' R C05 ei'R Sine :‘ Q a“! 9““ d‘flb-WGURE 9 Grid curves on the cylinder 09+ 0+“? Caimcler' 6. Consider the parametrization (I)(¢,6) = (R sin¢eos 6,Rsin ¢sin 3,}? cos ¢), 0 5 ¢ S It, 0 S (9 S 27:) for a sphere of radius R with equation x2 + y2 +22 2 R2. Compute T¢,T9, n(¢,6), and "n" . Tq) = {Erase Cord?) Rsiaemstpj -Rsinq9) z, -' (-Rsfnesincp, Rwasincp, N» at _. L J h- h(cp’9): To “fire : \QcosGCong Rsiitemscf ~Rsin¢0 ~42 9319 513‘? 1209599599 0 = {R case 51nd), R SinBSIBJ/W Rcosfil) 511159 —R sing? < (0:951:10?) SmSSmCP) (05¢) ’1'; UPCI‘DV 2 R5 .5;ch 6r “:9: TpYTe '5 MW"! FIGUR 13 ”h“ " H R SmCfléY-H -'-' RLSMCD :r’ngcp 1's Inward Paidiifl 696415? 7. Consider the parametrization CD(x, y): (x, y, g (x 32)) fora graph of a funct1onz—J‘ x ,).y Compute Tx,Ty, n(x,y) , )and "n“ . ,‘ .97.: _- _. bi _ T”‘(JJO)%*> ) \y— 49!,3,> thfl): T): X?) _ L, :5) k 0 I 8. Find the surface area of the part of the plane with parametrization @(u,v) = (1+ v,u — 2v,3 — Su + v) that is given by OSuSl,OSvSI . flu 3§a®:<01‘2“i> ) Fo=j¥ufi:(b'32‘7 _._> _. t 5 i: ‘ Tu *Tu : ('53) ":5 -" (-4) "5/47 “ll t GHVH-Slz’ +(.,)L =- 107 — d _. l | I 1 §UF¥O¢P Cut-ea " )0)“an H- “ ff‘hb? dudU : ‘ild7fiiufdu .2 ”I07 6 a 0 a 9. Find the surface area of the helicoids (or spiral ramp) with parametrization (D(u, v): (ucosv, usinv ,,v) OSu SLOSVSIE. Tu: {cosv Smu/ o> Tu: (—usinu ucaJuJI) 'L g» k :TkxTu' COJUS‘muo - - "‘ — U “MM Mas“ ( Smu) co! )u.> WK“— _ m_ m 1,3511% TQIDIQO‘FIn‘i‘Jm“ 5-19»me Surface 0119‘: : gllnlldnzdj‘ J‘Ofdli‘uidudv: T? [LiWJ'g-‘E‘ (”r-JD; 6r} byTrig. "SvloS‘i't'i'u‘l‘iom’ fit 14:: hang JWLGI'Qy‘MJ :E[fi+-Q/xli+ui)j 44‘9“ dx= 99c 6-016) W:SQCQ S U +1 ldu - ISecesec 9:19 23559: 939 - 'i‘lSGcGtme-iflml Sece++w§2k§m1+lngmE§ 10. Find the surface area of the part of the plane 2x+5y+z= 10 that hes 1n51de the cylinder 3:2 +y2= 9. :<l,o):1/_> )T 3:)<a’,\j—s) -.:. K A {a MW , ’a—L : (2,5,1) )jnu:\la+1s+g=d§5 o I -5 SM“ We“ ' X}? “““M 2 Si 030% —’ Jig Slum: El“ 3") X17 5? X1915 Kiwi? 11. Evaluate the surface integral 1 I 1+ x2 + y 2(13, where S IS the helicoid with parametrization ro‘—-- ¢(u,v)=(ucosv,usinv,v),OSu <1, 05v 57:. i M .. . —-« "- . TIL! ‘“ 3;, " ((91115J Emu/0)) T? —€i= (~a5mvlucosull) ¢\ A A “(UUJL'TXH‘FZ C03us‘uo . ’ “ ° m (Sine-cosu)u) )“nH: ”1+ Q’- -uSmU Man: I SJOJ'J'X +7 ”d5of:f‘l51+(umsUIL+[usihU-)* 1113 dkdu (fl/UH: Hulda”: a di‘éJlH-quu 31101139. 36.! : 'L‘J'filT 12. Evaluate the surface integral J]; xde, where S is the tn'angular region with vertiices (1,0, 0),(0, 2,0), and (0,0,2) g is +9.9 region In ‘W’ (3‘0“? 3X +7 +231 61‘ ‘2'-1~1K 7 over 'HIQ dbdham D: ffXHHO‘X <1 0<}'€2”1¥§ (11°19) EDI y)” 1”1X~7))0‘x‘l)onyI-lx 7 7:14.15: I¥2= 3(x,.,)=)22cy+leH 15 "‘“‘ ) n 1‘“: [1" '1' j] S: 5 1 2x 2‘ 8’HHJ1+(-Jlri—JL=U!E>I W ofxlwrld“ ; r Vfdrdx WE: if :24» ”UM-Wink): 33°[2V3x Ma]: WfJ-anzsg" 13 Evaluate the surface integral ”(x2 + y2 +z 2)d’S, where S IS the pat of the cylinder x2 + y2 =9 hen? the planes 2: 0 and Z: 2, together With its top and bottom disks. Let g. be [Iii-Nd Suffice S loo’rke Swfue 19(62): (3(058 39.136 %) 0560565254. 313 1,941,415 «51:th - -~ Fer 'Hnis Cél’lnal-Er 0‘? radio: 3/”1'1”: 3 55 I‘M-webs: J:f/q+2)-=’3d2c|9 39[17a+230f )"(rcose,rsi;el)lofrf3 0595.111- L " E “:TTXTG‘" C°5951;130):T ,56 ll —l"f0599(‘r0590 3 555$ (X+7H+2)ds -" 025 of(.,’- +4errd‘9: 0:“ if(r3+‘frdr)::filirfi+lf}3: L???“ On 53. :11§(3 r): (V3056- Y‘ng. 0)) O<r<3 0—59‘21‘1' 3,; o S (X +2 J=d5 f :- Qr, -BJ.7.- Homework #26 3 Rer-{Zad Sectiou 1'6. 4 xffli'o‘) Y‘ dr‘d 5 :jjT ‘90 l" d r $3) 3- (Due 04/28) Do Section 16 4 Exercises: 3, 5, 8, 9, 17, 19, 21, 23, 27, 29, 35, 40 556127;,2213: 121m + 13rd! H Prepare for next class session: Read Section 16.5 5: 5 15 +5) 2‘“ 1 ._ ’0’ ans; :lil'tn-K Ohslz $(9 nll=r ...
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