Math241_Fall08_2PM_Exam4_Solns

Math241_Fall08_2PM_Exam4_Solns - Math 241 — Exam 4 —...

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Unformatted text preview: Math 241 — Exam 4 — 2PM V1 December 8, 2008 55 points possible 10. _ No Hats or dark sunglasses. All hats are to be removed. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as .a time—piece. No music systems — lPODs, MP3 players, etc. ~— or calculators; same rules as with cell phones. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You wili not be permitted to leave the room and return during the exam. Every exam is worth a total of 55 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 6 pages. Be sure to print your proper name clearly and then write down the title of the dis— cussion section (Le. ED?) for which you are registered. If you finish early, quietly and respectfully get up and hand in your-exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready. Good luck. You have fifty minutes to complete the exam. Name: ____________________________ __ Section Registered In: ________ W 1. Consider a metal wire in the shape of the parabola y : 332, 1 S a: g 3. (a) (Spts) Find a parametrization of the wire. rfl\:(tfi@3) igeei _.—~ —_ _.——. .— (b) (6pts) The density at a point (3:, y) 011 the wire is given by flag, 3}) = E— g/cm. Assuming $9 g L7 2‘2 sf ll (i, gall At I A st is w... ism? , as: ea MN 3&5a 39 - that the units on the Lt—axis are in centimeters, compute the mass of the wire. __,__.__ (c) (2pts) If your parametrization in C was reoriented, how could that effect the integral computed in part (13)? Briefly explail your answer. (A ‘9; r C N ,, n hr- \N mlr M Q a@ lgj meslflbllcx M M A Ragga)" 1 in”: 5Q Ag My M066 gal/bi “9 W“ \{m/x lf/Ml UV‘Z: 9 an!- (a) (Spts) Compute / 2 + 2 , Where C is the circle of radius 1 centered at the origin, C 1: y _ oriented counterclockwise. deJ—scdy I X; +\l‘5 “ 5 @5215:— + Swee— <31“ : g met (5 :2 «mt-)1)“ écltli‘fi (c) 3pts) “{0de Green’s Theorem have been an appropriate alternative to evaluating the line integral in part (a)? Briefly explain your answer. Nb 1 fl:{>(p\\ ;(%5 ll 0“ (ll? Jisl XQWKLJ. (3pts State the formula at the conclusion of Green’s Theorem. ( \ Nat lenms 3. (3) (4pm) Let S be the Sp ere 51:2 + y2 + 32 z 16. Parametrize the Sphere. HECQAA : r (LlLo$b$m4/ Liam/yam}, 4509 ' 61; 9!: 9w) 9- C‘riT. - -_—_ {*4 Via; v1ch L-[CaSQS\/2J( l Eé Kgé >< ( 4:6, :5, A: ), 43w: £4334: f‘46m _ ‘ a __ #8 $3: 1-“— Ca (0363\09é‘ ) ‘— I bgmasm Cg ) (69-?EL/iabim4 54‘3§5\ I 4. Let S be the part of the cylinder $2 + z2 = 4 bounded by —1/2 3 y S 1/2. a ( . () @335: (class) «L‘ Jéww§\ ) 058.99% “are E 0). i3) Esta 51" S J 4pts)IFind a parametrization of S. . I (3pts) Find the general formula for the normal vector. 0, (905$ Q ) 11 c>\ 7”: (*Qtégg/ 0; ‘QS‘AS\ fix c 4pts) Let F(:r: y, : (ageing, 2:). Compute the flux of the vector field F across the $1 a.— ( ) surf ’FM -‘ ( 3% <‘ ~= Sit} [JZOSfi ,Sm‘h 96an ° ,JA I "I “T 8‘9 (9'4 egg—499253 b "[3! ‘ A r. “.4 ( Q’fl'\ (3" “SC\ _.— : we.— 5. i i 2 (a) (4pts) State the formula. at the conclusion of Stokes’ Theorem. , r“ A I A i g +°ASZ<§ (VXroig i es s i i (b) (7pts) Evaluate (\7 X F) - ndS, where F(m, y, z) = (652,42: w y,8251ny) and Where 3 S is the portion of the paraboloid z = 4 — :52 w y2 above the my—plane, oriented so that the unit noinral vectors point to the outside of the paraboloid. if ’L VC Cari“; 51A emim’h‘n H“ on 01 "kW/ti Fa o— o 3 .lg W “Lao—i ac inn OcSSiioLf‘ L We QLLZSi 1 m WC 3“ egfiav ( £959; 393w? I6Lfi—JW «We =gimme.o»2~aswwxc2 I I <:r[,.c33mé + H2586 P45464995: I /~Qema + }@(H[:aa\'4&’”§m6\‘ie i : (“99% FF «F 960331;» “leseéi/Pexcle £5 i l WMS : QCJSéiiq—r “L g6)?” 4- L/Slflab/ ‘ -._ O C) ., 0 +19? +0 "‘0 fl... épz gqstvxavng‘? I Math 241 — Exam 4 — 2PM V2 Name: ____________________________ __ December 8, 2008 Section Registered In: ________ __ 55 points possible 1. No Hats or dark sunglasses. All hats are to be removed. 2. All book bags are to be closed and placed in a. way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now. 3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time—piece. 4. No music systems A IPODs, MP3 players, etc. — or calculators; same rules as with cell phones. 5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam. 6. Every exam is worth a total of 55 points, Check to see that you have all of the pages. Including the cover sheet, each exam has 6 pages. 7. Be sure to print your proper name clearly and then write down the title of the dis- cussion section (Le. ED?) for which you are registered. 8. If you finish early, quietly and respectfully get up' and hand in your exam. You need to ShOW your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID. 9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready. 10. Good luck. You have fifty minutes to complete the exam. C 1. Consider a metal wire in the shape of the parabola y 2 3:2, 1 S a: S 4. A2454 (a) (3pts) Find a parametrization of the wire. (b) (6pts) The density at a point (:13, y) on the wire is given by f(33, y) = w: g/crn. Assuming that the units on the m—axis are in centimeters, compute the mass of the Wire. gov/1 M wSlam l ( ( Wag {63A (0) (2pts) If your parametrization in C’ was reoriented, how could that effect the integral computed in part (b)? Briefly explain your answer. 2. (a) (5pts) Compute / 0 5324-92 oriented counterclockwise. . E I m [ad A 0A v69 0" , where C’ is the circle of radius 1 centered at the origin, 2:er (b) (3pts) State the formula at the conclusion of Green’s Theorem. (0) (3pts) Would Green’s Theorem have been an appropriate alternative to evaluating the line integral in part (a)? Briefly explain your answer. 3. (a) (4pts) Let S be the sphere $2 + y2 + 2:2 = 4. Parametrize the sphere. Eflaflx :: @3633’4} oygi‘néémch 059,49? ) bé—éf—TI‘ (b) (Tpts) Evaluate // (3:1:2 + 3112 + 322) d8. S (Gama haw—EDDIE. EYE—£4: 35xn¢£[é,¢\ HEMZ‘EAFH: 99¢ (:93 tile/14). gggexigeasug; egggeyaexgg ——~—-— ggg 4 manage m g 9T?” [were (we 5 w 86 O ‘_ f; g, {—5545 a gym; QT c; - 3 WM - “)T o 4. Let S be the part of the cylinder y2 + 22 : 4 bounded by —1/2 3 a: S 1/2. (a. >( _ = : \(g,4\:(g)9%i,;stt\ wee/4, may. _ 4pts) Find a parametrization of S. (b) ( ES\LE{3X( \; O; (A ( b, .994, 010093 «-— [o) #9031, "ag'n‘L\ E a 3 a § E s 3pts) Find the general formula for the normal vector. E ' (o) (4pts) Let F(:c,y, z) : (sinx, z, Compute the flux of the vector field F across the La Z“ V , z 8%; (6M (El ggmrt Iggy“ t. (O) raqu—Qsm-L\ €15 AL _ c) *’/ _ _ a: g V: [F4 Sin—L(as-L -— Clfi J/a _ figgj‘r ( gun—tagg3r\c1<; 1 A CSQQ'W émjtcasjr 3 ’3 I 9T i :- 6» I t I t I 7" “(K glna’E a b 5. (3.) (413133) State the formula at the conclusion of Stokes’ Theorem. 4’2. Cl :4 of 4. 6 2 (W90 g - ES 3 (b) (Tpts) Evaluate (V X F) - ndS, where F(x,y, z) 2 (($22,433- — y,825111y) and Where S . S is the hemisphere z = M4 — 392 — 3;? above the xy»plane, oriented so that the unit normal ve tors point to the outside of the hemisphere. a a gblésTFIVt/r ) OCH/(S M %;O/ K+YE L) . a ‘ I QCO‘D‘L *J§A+) (be [“ QS’I’I‘é, Jaos’é, (B . r f: (*ngjr + “afloat "ZléijfloS‘L\ \tn- I... 0A Vera“ ...
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Math241_Fall08_2PM_Exam4_Solns - Math 241 — Exam 4 —...

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