EXAM 3 PORTION - Math 200, Spring 2010 ; Name:__KQ_..____...

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Unformatted text preview: Math 200, Spring 2010 ; Name:__KQ_..____ Exam 3 — Out-of-class portion (60 points) Show your work or give clear explanations of how you got your answers. Credit will be lost if work is incorrect or does not support the answer. (6 pts. each) 1. A particle starts at the point (3,0) , moves along the x—axis to (—3,0) and then along the semicircle y = J9 —x2 to the starting point. Find the work done by the force F(x, y) = 24xi+(8x3 + 2439/2) j . C wrg‘fi-a: : —j'§-a? = a} away + (3X?+1‘lx7l)d7 “C -C. - 3 ‘0 Gm, 3 ‘3‘} : — $er i'idfix *3¥Xyl)‘ib‘fxijdn Zrib-Fever:- —- C kud- 91:614.»)? __ _- 1 1 _ fl' 32‘ 1T 3 “was.” ~ on“ “a, )dn- ~19} jrmdrde= ~>s[e)o.[g;rw :—'18£1r 00 O 2. A particle moves along lines segments from initial point (0,0,2) to terminal point (0,3,0) (as shown in the figure at the right) [under the influence of the force field F(x, y,z) = ycosxyi+ xcosxyj—sin zk . Find the work done. 3L i .5 z .—.I i 57% g-z 7: (0’0, (05 Xy-Xy S“~{7-[(OSXy-x S‘AX) {0.0.2} CuriiFi- 7035):, 'xcgsx7_gmg _ 7 7 --—_3 ‘- <0/010>: o —_‘ 50 F is Conserwiiue Fri“! Ioaieniiad Q9 (Abe): 0L2 fi‘ikymxy 3 ‘?(§yjl)=J7C°3-K)rdx 1 Snitx7+5ly1el (3,0,0) x A _ . . on afP—Ywfly . 537bme +3019) = xcm7+3r :xcaigégyr—O ,S°%C7,%l= Me) A _ - . _ UM Aicp“"g”\% - g2 {SMXy +k(%)) = 1oz =-S'm2 1') his; fin—Ed; : (OSE-TC- flfr‘eg‘r? (pi ijlc) = Si”? 1- cos-2 ‘1" C. _ (0,3,0) may w ” S Ea}: '~' $7513.01? -_ @(cfifipl «page» = sako+coso-l.§{no +w52'1 (0,0,2) (0,0,2) = 1 —cas 3. 3. Find the mass of the cone shaped surface 2 = sz + y2 that lies between the planes 2 = 0 and z = 2 , where the density of the surface is proportional to the distance above the Jay-plane. faramei'rike SUT'FI-"acoS-Ots firqfk 0‘? 2-: 3{x17):m §( -_— < L 3" 0—. _ my) ( K) 7/ “1+7”; 0' p“? ‘ H >" " (—35-37?) :<"\F:“=- yl- if; 2 J > . 1- Y1. ohms-fig F6953) = [<2 = sir—m7?- )m” : \lmfifi 1’ is}... ‘ xix,» + xfiyL-f‘ 2 Q1 m 3 fb‘malds '-'~ K \ Xl-kchi 5 .5 a 21 1 ‘ Mi W‘w‘” 6‘: an = Kfij' amt-ole : MB. (in) Rf)" = LEV“ 01X+yafil1t o o 0 3 4. Find the surface area of the torus (doughnut) given by (13(u,v) = ((2 + cosu)cosv,(2 + cosu)sinv,sinu) , 09522130932”. , @(u’u) :((2+ cw“) (05”: {2+C°SH)SI‘nU) SCH“), 05. ui‘QTi" O f UJJIT‘ Tu. 3 <~SmuC05U , " Slim S|anyCOSUL> TU : ( -(1+Cosu)s{nfiu, (21+ tests) (asu,07 n h... '-J’ N I l “:1 Tux U : _ k - J h k “SmuCosv -Slnu-$lh\) cosuc : <_,(1+C°Ju)ceju_C0fU , “(1+tcsu) sinu (3+:asu)agu 0 -(1+(g5u) cosuSCnU, -— _('1+Cosu\ Si'n u > “Ex” 3 WWW ' (lthSu) C05 QEGW‘i’Sm J = \S (2 +(05LQI 5' ll-Hosul V +61 +035 ls): Sihlu Swine ZIJ“ 2"- 2“ -'= D‘s-(Mu. zrr ‘ " are“ {1+cc5u)°[u_olu : ujd ‘[2u.+snnu]o = Qfi{‘+n) : ‘3??? ° 6 5. A fluid has density 1500 and velocity field v(x, y, z) = (-y, x, 22). Find the rate of flow (flux) outward throughthesphere x2+y2+22=25. CMU(U=*) _-_- a); by) 4-5:”) 4‘; (2-2) :2 canSl'ld 7 rate HT: 3"“ 3 :Isoo ‘3 Du: __ g Q J ‘ liux ' U ~d5 d\‘\! (UldU mineraeme Water») 90M: .5 =f5jgldv=2juu ‘e r—5 3 = 2(isoo)(§ TTS ) = 500, 0001? 6. (a) Can you use the Divergence Theorem to evaluate the flux integral of F(x, y,z) = (5+xy)i+zj+ yzk across the surface S that is the 2x 2 square plate in the yz-plane centered at the origin, oriented in the positive x—direction‘? Why or why not? No; Hui Gideraehee TLQO'I‘Qn-s Com be used 'h 4K4 +LQ -Plux over 0» boon-Jana. Sue-Pace. :3? CL. solid region / no‘l’ over q Planar realm (b) Can you use Stokes’ Theorem to compute the line integral {C(in + 2 yj + 22k) - ds where C is the straight line from the point (1, 2, 3) to the point (4, S, 6)? Why or why not? No, S‘l‘okes Tln€onn~ Com enha Ira-e Mail on 0- Gib-Wet bovmluna CON-’9) Ml— Ocht. lib-Q, ngm-ent. 7. Use Stokes’ Theorem to find the circulation of F(x,y,z) = (xy, 22,3 y) around the curve C that is the intersection of the plane x+ z = 5 and the cylinder x2 + y2 = 9 oriented counterclockwise as viewed from above. c.“rCu\n+‘;“n :Aéss F ~43 1‘ CUr-l 'clfi-g (Shh; TbPN'Pm) .1 L— E CUP‘(F : ) {5: £3: = Mango-x) : (Mg—x) *7 22 37 A '\ flame: @(X : 1 ti: \LI OH = 4 H) (X’y’g'vi “71-”? 0 l 0 1,0,1), Ufwamd Paid-$3 norm: git-.45 : gem! (ENE : JD Curl(fi).xdfl 155% (Horxlfllop an ~ 35 (Mn: yr 3 “or: c. D ‘ “M We 1 t2: emits? : (summers-mg; r. 3% ]- - : a an o o ‘irr 8. Use the Divergence Theorem to find the flux of F(x,y,z) =(x3 +tanyz)i+(y3 we” )j+(3z+x3 )k across the cylinder bounded by x2 + y2 = 4, z = 0, and z = 3 and is oriented outward. S'iux 1' 1 Iffwdio (EMU (otheraence The.er - .——* _ _>_ 3 3 1 Awh‘) “ 3X(Xfi+any%)+3}(73—exi)+§i (32 4-)(3) 7373+}? 4-3 Lu : E 4 ‘ Suit?) c Ila-c1? u'ri-i. Mlbi l ( rI6)%)lO~v-2, 02912.net an; 05;? M54 3. $35 “15.0st = an (FM '~' {QT/hr“) r “W” 6 : @52W57&3+r)d%4r39 0 big in .. 2-33 “ “A (rawflej arde o o 330 2i? 2 3 9 d9 ' 5(r3+r)dr o b 9. Use Stokes’ Theorem to evaluate the flourllT -dS , where F(x, y,z) = xzyzi+ yz2 j + z3exyk , where S is the S part of the sphere x2 + y2 +z2 = 5 that lies above the plane 2 =1 , and S is oriented upward. - 6’ Sh keg, Theorem.) (Uri (? )a‘s’ : § Ed: 2%“?! (F7015 " 1 m - ' 1 l l * —’ 1‘9'6 S: is disk. x+y 5-H) 2“ amides! UrIUQr-cl mil-k “Mme-l n“? S“ skews HM; Sam,“ bobnd”: “Hg 5 J Hm Par-i of” ‘He 59%? ¥:y’+2::$ than: ' ad . . P = Cur-i : S 3- i % g __ 7 xy 2. 3x a; a —<x%e~ 3"" ' x‘n 72" ezexr A ‘ in’x’wvie) “3) Curl(F)- K 1' -X1E on 5| usherezzg 1 S; r J "4‘" 4 "“ 3c” NF) 45 -j;ChHF)‘dS :1SIC»I-\(E).de “79511:: 1-5—1 _j2511~a. m ‘3 fin ‘i X $45 - j- \- coser ralrale : jCoSI6J65“r dr far)?“ 0 o 1 o 6 2 : JL _ .L “i _ — _. 05 2 (HCoflQMG ., r 10 _ { (2n)(-:})(15) — — an. 10. Suppose the components of vector field F have continuous second partial derivatives and S is the boundary surface of a simple solid region. Show that “curlF «is = 0. 5 Fa $m‘e' F : <F‘)fi'/ (:3 > had (Oldie-woos $e£enel {)Q‘A‘l‘al der‘.V‘¥+1‘“9-’ m“! S is 'H\e bonndahg Sari-ace ci- 0. Slimpie Solid region {call N) 9 “fl Ca“ v51 We Dfitt‘raence, Theorem SiCUrH-fiydg 3' jfw Ch.“ ((url?)dU as 3; =5] \U' '1. [a§,_gi+§i_a‘s a"; A‘F 3):; 3x33: 2732 3753’ 3231—fi7]du ‘i‘erms Cause! {A path-s icy Clair-nut; Theorem. :jflwosuro ...
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EXAM 3 PORTION - Math 200, Spring 2010 ; Name:__KQ_..____...

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