HW3_solutions - MATH 2401 Due in class on Sections F4 85 F5...

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Unformatted text preview: MATH 2401 Due in class on November 11, 2009 Sections F4 85 F5 Homework Assignment 3 Name: gtID#: Note: There are 10 problems in this assignment, and six of them will be graded (you won’t know which ones beforehand). Write out your solutions neatly and explain your work. Please add extra sheets if you need more space, but do not use the back sides of the pages. Problem number _ - _ _ - —‘-_ - CT Problem 1. Let a > 0 be a constant. Find the average value of the func ion flay) = $2 + y2 (a) on the square ——a S m S a, ——a S y _<_ a. firm 1" (04%))”: 0‘ 53f : gag“ UL—bfil)okxd\3 o o q q = flaw-MM S a Q 5:" \ ----- ’23:} a = $38 ildxo‘tafi 0 0 q 7, 8 O\ “ESL 7" :3"? 0‘” 73> he:ve swf‘ifiowrk :2 go‘h [ML : (b) 0n the disk $2+y2 gaZ. M 2 /\\"o;7“ USE Tuber Qaad‘ck ". 24% q Rng akdi : 3 g 9* we 0(9- 0 SS fish 0 a A /\r L‘ v 3 V /\ 0‘ .. ‘ 5‘ _, r ok- : lu —-— — vQ/ngD 4 2, .52 r? _, RNWQSL —> 2 ma» {f Problem 2. A plate occupies the triangular region with vertices (0, 0)7 (0, 1), (2, 1). The mass density is given by the function /\(ac, y) = a: + 3/. Find a) the mass of the plate. ( 2M3: : 8‘ SL3 OW Chg O Mm» \ ( mmwmmmflmmmwmmm‘wxr (/"x‘ “am. 3% Problem 3. Determine the triple integral fffv zdxdydz Where V is the region bounded above by the sphere x2 + y2 + 22 = 1 and below by the cone z = «$2 + yz. Determine the integral by using 7. (a) Cylindrical coordinates. *Z_\_KAL_\_%'1: \ :3 r +%7‘ 2 %: 5:51:33.“ :5 2" “x. W “:AJrerschmw «=> flag) +08% \Q”§=\ t g7 r z. W? W S$d¥dsfld%=g .g L g %\‘ obcdvak-G V VG 0 $11 ‘- Vi? ‘ I r— f Vt o\ :IU'S r (\PV1)~T11 dr- :: (1W 8 V O L Xmmmmfimwmmfimmm((anxmmnmmw‘mwu‘wmmwm A - A A f2“ ‘ L Problem 4. A homogeneous punctured sphere with inner radius a and outer radius 6 rotates around an axis that touches its outer surface. Determine the moment of inertia. [Hintz Use the parallel axis theorem] first die/WKM \MWWLVV’V of: ran/w GEoW*'QWLQX\S (J'hmufik W canker ()5; W3 \ :«CMMuS’O-M Z—AYL‘t “Wm/\- - \‘(Xcfit‘Zi3'3 3*??? FL . :1; A 15 W confirm/Vi Mmsxfi'wv‘ M W\ -32. IN : AWELI LWFD?‘ ’ Li :2, <3 mmomfim+ oi; \VW-“H‘” \5 WWWWWWKMM‘WWWWWMfiW})mmmwm’mfimmavennxmwkxmww—wmmm:qawwammwmflnv‘rem“ ./‘“'“\e Problem 5. Let R be the solid enclosed by the planes as = 0, y = 0, z = 2 and the surface z = 3:2 + y2, where x 2 O, y 2 0. Compute ///R xdxdydz .7” 4’; {Liz-r oxvud 7:”:— (n‘l‘besLC‘F \n I r: 57? 50 417 \S'? a. “Mafia-4b : S g g Toosg- r [email protected] E o 0 V?- X 4%. (3:: L :8 r1 g (1% ORV c> 0“ r1 W L L : ENVG'B 1% rL (l—rydr 0 o W __ Zr}, F“ 0“— o S W ” - ELM-3 ' 3 S d 5/7. 5/L 2 3/1— "LID—[L __ 2‘ —— 25’s ___ 1?; 9. — 5 —* 3 ._. 5/1 5’3 ” 9 8‘5 ,rflx Problem 6. Find the centroid of the part of the sphere x2 + y2 + 22 = R2 that lies & Mk mm a) above the plane z = c . Here 0 is a constant, —R < c < R. We COM assvxwu, (W93? Q20 CoWw§fi rgfif’bua S»?\4\JA"Q E2 S$MW,M\/m\x~g 2' O ‘ \Ne use cmhadr‘mad caofixmoA—m ; 7. XZJc‘Qlttt 3Q?" ==~E> VL+°LL=P~ wtn‘tENSefilrLflm M 72:52:17: \s vax rl+CLtR1 “g r: {L__C2 N ,. S machete- O O W»? r we 0 Z Etc V ’Ql—rm ——CV‘ 0“, 9 3/). L EL’C’L (Eel—r} L ,Q/w .o ——c L 3 0% Cf Rl—CL R __ ,_ .___.. —— C + 3 ‘* 21“, :7; Q. Problem 6 continued... 1 i 'L Aka, “2/4" E “C?” RE "i EV: ‘ a“ a c W W g13 d? T C, = 19‘ g 0 [m —w(; v < Lbrlfi- \ VLS I 1 «LL Vux a (53 ’" \J w ‘i‘ = . W3 m «s cm idem? :5;— CRJVC‘) (22" Cl 3 8 R“ + 3562+ C3 3 r 4 a??? a: 03+ C3 fl hhhhh i CRJcc ) 0271 C7") @‘h’o‘z‘ 3 (0/ C / 4 QEfL-ECCE—EC) B WWW§WWWWmmummmwmwwmmmmwWNWWm“MNWWW‘WWMWWWW\wmmxmwwm‘WMX f”? Problem 7. Let Q be the region in the yz—plane given by 0 S y g 2 and 0 g 2 g (y — 1)2 and let T be the solid obtained by rotating 9 about the z—axis. (a) Find the volume of the solid T. , / /' / / :5? . \ m» WV Tm; o\w\"\,ovh M 3&0)" .‘3 <§~L4®WQ€QA Log \M'l" (\Q-v l) \. i 02%2Cr?\31' r—fb :3 tan“ Al r &T’ 20‘ ’hxxm‘l/S Gwlfi iv‘ W —~§=lqwe 516m». / u x t 1 K Problem 7 continued... I l ,5 (3. \ _ ‘ a c; <4 (b) Find the center of mass of the solid T given that the mass is uniformly dis- tributed. £ :15? 2:: O , N30 :52} swam. “\fié , xv“ W '1, 3 %’ )fl(¥\vji%> d7: M , fi/ (VA L W 'L 1: 3 g g :2: T‘ a\% d V O O C) (r _ ‘31 O O '7_ re ‘5 e “_f}_ '5 I} :yfif’é-Wn“ 4.) *e— a Afi— 15' 9i 15+€>L1*%‘X+ (L r ‘55— 5 \“\——/ 2 \3 ‘ 5 + v)— 3 \B - AW U5" ééu’ : m4? 3 i _. ALL/L“ ' 5 \ ’(\ AMT/5 "KW/5 , 15/ 5 : {-56— __ z: \ %M~ m 3 / ’5 C’Wwéfirwss (0‘ or?) 7.7:?“ 10 ..._./ YWWWWWWWWWWWNWWWWMWWVWHQVHWWC‘VM‘QW» Problem 8. Find the centroid of the bounded region T determined by the inequal— ities x2+y2—22 2 1, :r:2+y2 S 5 and 22 0. Use Cb\\ndfiCo\\ AWL“ ‘r '2._ 2,__ 'L :> 7-__ %12>, "fl/,w \A—kvi‘ % >,—\ V l ‘ :5 {flat .4. S :5) r?” e S Q T 11 Problem 8 continued... 12 Problem 9. The volume of a solid is given by the repeated integral ,.., 1 x/l—mé x/2—zz—y2 ( dzd dzc. \ f / / 3! v 0 0 W Compute this volume using spherical coordinates. % —; {zigfii Fags Q fg‘g'fiQ CDS a? 2: 9,3an '13:? m. «v f; /4 15M? (if A (Q 0L9- 30 30 mg 4% 6" 16$ fl“ smcfidcpgdfif { ° 4m 3 G? J: [mde @327) 2 O «\[1’4‘2 9.??? :75. - Q_ g :%Czfi‘i‘~fl= T§(W“D 2.5% Problem 10. Find the area of the region in the first quadrant of the plane R2 bounded by the curves y 2 x, y = 45c, my 2 1, :cy = 2. bit (L’s—3;: owed V=XLA 50 MW (\Alflynswafly/ \4Ulé—Lb) New 5 2 \Ne Mam Uv:\3 amok so x: \j/"‘ “Mi firm ‘ 1; H1. WV 6“ 2va 538/3“ “ 7%: Twain“: 3M0?” éy/V aj/AN ‘ mam” A: 14 Wmmmmmmmfimmmmmmammuwmwmxmfimmwmmm ...
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This note was uploaded on 05/04/2011 for the course MATH 2401 taught by Professor Morley during the Spring '08 term at Georgia Tech.

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HW3_solutions - MATH 2401 Due in class on Sections F4 85 F5...

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