Old_Final_Solutions

# Old_Final_Solutions - (1 Determine the surface area of the...

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Unformatted text preview: (1) Determine the surface area of the paraboloid O<z<1 \$2+292=2z, H by Whatever means you wish. InsFimaQ bu mekri it co—orols J we. ca l'lﬂéqrfct‘ 'Hnl. Sta/“ghee. '0': x = u. @090!) u com) (J = (A Sin (V) L3,) l"(q,v) : u {in IV) I ‘2. 2 = ?“ %u.1' an» osusdﬁ am osv\$2w C050!) - M an (v) -u?' COSUI) F“ x” : SHAW) X M. 6080/) = -— 03 Sin(V) M O u so 5‘5 “1‘ “#CaZM'I-UHI'A‘W) * U2 = “blue-H . A]? 211’ Mel 1. AM =5 1 as =j j uﬂlw.‘ 4mm II N 0 615:05! I 3-1—5 dt=2UAUk 3 l 211- '3’ “'3' 3 ‘ ~43. (2) Calculate the integral fa? dA R where R is the region below: (3) State the Fundamental Theorem for Line integrals 61M 4 gum—hon ¥ Mine} 0» 1‘“.le (or in “Whe-Space)‘ owﬂ 0. Curve. C £00m 1:, +0 1' Hana (4)+(5) Compute both sides of the Divergence Theorem for the cylinder \$2+y2gl 03231 with F = zsin(:122 + y2)k. (Of course, they should turn out to be equal). [5 v.5 ow 455-43 [LE V-f == Sin(x"+ﬂ") QM) I l 7"“. . 1. 01?; “Is: So L L s;n(r) rAeAr l I -:: 7. i (r1 alrdi' f° f" ﬁrs“ ) l t=rl 9H: .1er!- = l; l: 1rs'm(+) a“ 4% :- Jc: TrEI-COCCUJ 01% = TrD —cos(ll:l W Th" Swlaa ’9' cylinder Comfrises 'l'Lf‘ee pieces: [Extra space for (4)+(5)] 06W) vuSiMU) o = umstv) C TOP 2;: “sh (v) 1"“fo : (Siam) X (Mung): ( o ) a = 1. Coﬂh‘chu‘l‘ioh = J: jou— (E‘MJ’). (2) Judy ' - .. ‘ uh ion 2:- 50 Z11“ us.n(u2)olu = TTD 050)] i‘ws Bo‘uim 2% Since, 2:0 9 f=o W great Con‘l’nlow‘ts zero since 5 Paws ale-3 44¢. Sui-Face. M" “Wm'ﬂl-ife AVE-“=0. Condw'coni RRS = TrD-cosclﬂ + o +- o -.-.- «Eu—msm] = LHS. (6) Determine [de R Where R is the intersection of the cone 3:2 + y2 S 22 and the unit ball \$2 + y2 + z2 3 1. Pic‘l‘w‘c s PMKH 11* ‘11; 21:! Calincil‘mel Sjmm‘i‘l‘j "T—b OSGSZTF. i‘AV :: '"7 211‘ I MM £430 j; gash?) ngiMc?) A3019ch 1'7 “‘74 = \$1.." j mshﬁsinue) ale? 3 :3; f6 Sm(2cf)olcp = I ‘3 11‘ m 20W = = iﬁ'f swuemcp = -1 barb” Pri- as about. 3% 2 (7) Justify the following statement: If V x F = 0 then %F-dr=0 .1 for any closed loop 7. Sol”; 4H ?‘TI is Maul Mum (or mom. and-t“: on a sih‘oln Gonn¢¢+¢A Mel fo =. 0 HM,“ E = 17.? 3'0!" Sow-L 't’uM-‘H 0n 815+ 'HNLM 511 +L... Morena g1"- L (M In'\'earo‘s ) * Ac‘l‘uqu‘d’ gor- Sud» c. Sui-Law. 4b add/'1. "4' 59.92;“; +ka+ F In. in It S‘mlo’U—cannu+¢& (tron: ab:‘i+a +9 an¥mc+ a. looP +0 4‘. Pom'l’ i5 czuEVA'u‘i" .10 ﬁg, Possiblﬂo Av ‘lean M at hob.’ w2H 1 wk“. (8) Consider the region R given by 032:: (y_‘r2)27 Use the change of variables \$=U, yzv+u2, toevaluate / dV Jay—\$2 l O O jampID'an '= z I zu l o = V Charla he. 1. O 1vw v giuq‘l‘ions MMMJ 093mm an new co-orae's: 2. Z 'L 7. ongsv I: uSv-m SH (Le. oswsl 3 osv<u—u‘- not. GNU-v3" hurt;er 05. “SI (32-. TDFC"WI'€. «have... 2. I u—u | v1 Ans = S S- j clutch/elm 0 o a V (9) Let R be the region where 0 S y 3 cc and 522 + 342 2 1. Evaluate / dA R (5'32 + 92? by switching to polar coordinates. ._ o | F4 1‘74 l - 0° :Jo :fl’zl 0'9 17,, = ID {—49 ‘n" H l (10) Which of the following is conservative: F:mi+eyj+aceyk or i F2: yemi+exj+zk Write itasz. c '7' k xF = V~L as; 353 752; :6C¢3-o)£+ o.. 1 e9 'xg'd 75 o a ,1 k foz =: 35‘ 3/33 \$6! = (o -O)c + (O '0); + (61- ﬁt '36:: C: E 5 IS M #Wﬁﬁm mean) E ;S mam-hm Now M \$043k #3 As é=ﬂe ) £=~de +3(:1,2) bu++k¢h ezf—“ﬁ = Eta-t- : So 23:5 L ¥=IJ¢K+M£9 new 2 -.= 2% = o +A’m So “take a Na) = {-z‘ tC, We M (1105+ Wat o+¢ﬂ+tal Mon IS 2 3 P (1 1) Evaluate / 005(7m3) d2: + a: dy C where C is the curve P... Cos (Tl-X) (12) Determine / d8 51:2 + Z2 over the oblique cone parameterized by 55 =v+vcos(u), y =vsin(’u), z = v for u E [0,27r] and v 6 [0,1]. '"V Sid“) l 1- Coscu) v (.0300 {a x 5, -.-.— V Costa) X SMOA) == vsznuo 0 1— -vL'l+ mam] 5o "MD": «I v" + V‘L‘stwﬂz = v. ,J 2. 1- Zunun cos‘w) Aka) "3 W = v D+aquﬂz+f 2V 2.?2605091'Cﬂszf“) Mn; 21' l Any-3.); 1 AVA“ 1'” 2T 0 (13) Compute the following integral by reversing the order ...
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Old_Final_Solutions - (1 Determine the surface area of the...

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