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# CALCULUS IV MTH 254

PSU

GPA 3.7

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This 29 page Class Notes was uploaded by Berta McClure on Tuesday September 1, 2015. The Class Notes belongs to MTH 254 at Portland State University taught by Stephen Strand in Fall. Since its upload, it has received 29 views. For similar materials see /class/168215/mth-254-portland-state-university in Mathematics (M) at Portland State University.

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Math 254 Practice Problems Wed Nov 30 Justify your reasoning Div and Curl Is y 2myz 1 z 22quotj y fin 2 conservative For Fyz myzyvz Calculate div F121 and curl F121 and interpret Ca1 culate djvcur F for a general point 0 The Fundamental Theorem Verify that 150531 2mm m2 1j is conservanive and use the fundamental theorem to evaluate dF where C is a pieceWise smooLh curve from 14 10 12 9 Find a potential function f for 112my 2 3I cos y39Z em sinyj 212 If C is given by Ht 2 165 sin 10 7T2j 1 201 O S 39I S 1 then evaluate F 177 by the method 2 of your ck osing 1 w 2 2K 2K b Dead F lt q lti 2l 39 2 HZij 27 Egt U i Qmwu t m 2 dw F j 1 I y Av FUJI3932 LI u zmwu 43 Cu flt0 0 Ky 0 O XE7 cwf fO J lt0 72gt m a mama WWOXX O 63221wa 3 9x37 2 ribarj axw m KXj 1223 aaal K Said foul HM O f B 0 Hxj acesj lt1 E J39 K 03 ltO172H7 0541le quot7 Sam s gm mm 0 new 02 ov 01 If m ltlt 1 71gt Mum r7 mltl1r2gt 50lltcfaij quotX Mz lt lfry9 K F CtCOS 63 s39n 4e 5 lt 1 x W W mm m m C Far ziltetws6l e sm r l 772 I ggtof easvtgt emarm HA 8 q AIL as a 110 an Mum Math 254 Practice Problems Wed Oct 19 Justify your reasoning The Gradient and the Chain Rule quot 9 1 Let fzy24y L a Sketch the level curve fmy 8 7 4quotle F7 3911 l 1 Calculate and sketch Vfwtl X4065 a I ax5 a c Find and sketch the tangent line t6 the level curve at that point Use a vector parametrization of the line d Find the equation of the tangent plane to f 139 y at 21 0 Of 39837 91 1 5 7 C theamd a 11504 Le Vltquot3 7 W V 7km Vim 39 quot7 90617 lt1 39 339 xSjO 4 WM r 9 HitB toldan wig 3M m dumd now Let V 1 I L W ta 1 I t1 I7 2 Batman is rying to escape a burning building He must follow a path given by Ft 2 sin te any in meters i in minutes If the temperature in degrees Centigrade at any point in the building is given by Tcy 101 y 50 nd the rate the temperature is changing over time when t 7r 6 minutes Do not normalize the tangent vector F t V1 39 ltwlquotogt W emchmJ 4 CchYQ loafo JH39 1470 VT f H QOcasf quotOat 972qu new m v Mquot be lt4 9 cg 7 TW LT Wm gm r6 8 lgt 39 K39zry39l 392 O H39CxQH 33 0 2 3 Math 254 Practice Problems Wed Nov 30 Justify your reasoning H 5 00 r 03 U 5 00 3 Div and Curl ls Fzy z 2xy2 1i z zzz l y 7 z2y2 2l conservative For Fzyz ltzyzyzgt calculate div 131727 1 and curl F121 and interpret Cal culate divcurl for a general point The Fundamental Theorem Verify that szi 2 7 yj is conservative7 and use the fundamental theorem to evaluate f0 F d where C is a piece wise smooth curve from 14 to 12 Find a potential function f for Fzy em cos yi 7 6m sinyj 21 If C is given by z 770 ti sin 1t 7r2j 1 7 20k 0 S t S 17 then evaluate fCF dey the method of your choosing Vector Line Integrals TF If F is a gradient vector eld then the line integral of F along every curve is zero TF If C is the path from 000 7 100 7 110 7 0007 then f0 VfdF 0 Let F 13 1i Bu2 1j and C be the upper semicircle going from 00 to 20 Option 1 Find a parametrization for C and setup the integral fob dF You should be able to come up with a parametrization for C in terms of sin7 cos Option 2 Verify that F is a gradient vector eld Then nd a potential function f and apply the fundamental theorem Option 3 Can you think of a third way to solve this problem Scalar Line Integrals Find the mass of a wire that goes from the origin to 11 along a straight line7 and then back to the origin along y 2 and with density function pzy z kgm F2t lt1 7 t 1 7 02gt may be a useful parametrization Does the orientation of a curve the direction the curve is traversed with increasing it affect a scalar line integral You should think about the structure of the line integral Also7 compare your answer from 1 by using F2t 1571 H H H D H 9 H Cf H 03 H 5 Change of Variables Let 7 be the region bounded the lines z y la 7y 717zy 3z 7y 1 Using an appropriate change of variables7 evaluate 4Q y sinx 7 y dA Don t forget the R Jacobitm Using z psin cos 67y psin gtsint l7 z pcos b prove that the Jacobian for switching a triple integral to Spherical coordinates is p2 sin b Using z rcos 97y rsin z z prove that the Jacobian for switching a triple integral to Cylindrical coordinates is 3525 7quot Triple Integrals in Spherical Coordinates Sketch the region of integration7 and convert to spherical coordinates 1 W m 1 dydzdr m0 gym gym z 12 22 Triple Integrals Find the mass of the solid bounded by the surface 2 9 7 x2 7 12 in the rst octant Lg2 2 0 with density function pxyz 312 Set up the integral with at least 3 different orders of integration Iterated Integrals in Polar Coordinates TF fol ff sinxey dydr fol sinx dz ff 6y dy TF If the temperature on a plate is given by Ty 300 my Kelvins my in rneters7 then at the point 17 0 in the direction 171 the temperature is changing at a rate of 601 Kelvinsrneter 1 TF For any function fzy7 ay dy 0 0 18 By changing to polar coordinates find the volume under the surfaee fry my over the intersection of the regions 1 g x7 y g 4 a d 0 g y g ac Sketch the region Iterated Integrals over General Regions 19 Calculate the volume under the surface facy y over the region bounded by a 4732 in the rst quadrant Sketch the region The temperature on a triangular plate is given by Ta y 1205 degrees Celsius Find the aaeraae temperature on the plate if it has vertices 320030 to o Iterated Integrals 21 Calculate r 22 dydas where R 02 gtlt 04 R 1 A A 1 Verify that xzysdydm xzysdasdy a 2 2 a 22 Double Integrals Over Reotangles 23 Estimate the volume of the solid beneath facy a 2p2 on R 02 gtlt 04 using 771 n wi a the lower rightrhand corners as sample points and b the Midpoint Rule 24 Estimate the average value of the function using part b Lagrange Multipliers 25 What is the argest openrtop box that can be made with 12m2 ofmaterial Vlw h lwh 12 21h 2wh lw By setting VV AVS Where in this example we think of D 03 31 H 30 31 Sl7 w7 h as the surface area function at the level curve S 12 This gives us the system7 wh A2h w lh A2h 1 lw 2Al w 12 2lh 2wh lw Lagrange Multipliers give us a powerful tool for nding absolute extrema in the case when the boundary7 is not a rectangle First7 set the grad equal to zero to nd all local extrema inside the region Then use Lagrange multipliers to nd the maximum and minimum values of the function where the boundary is your constraint ay 2x2 312 7 4x 7 5 on the closed disk 2 y2 g 16 Here we can think of the constraint as gy 2 y2 at the level curve 2 16 Local amp Absolute Extrema Find the critical points of ay 3 y3 7 12zy and determine if they are maxima7 minima7 or saddle points absolute extrema7 consider the boundary of a closed set in R2 Just like in Calc 17 we must also check what happens along the boundary Find the local extrema using the methods above See if you can also check the boundaries of the region ay sinxy7 on the closed region 0 g x g 7T7 0 g y g 1 Directional Derivatives Find the directional derivative ofthe function at the given point Dont forget to normalize the direction vector 17 ie make it a unit vector Then nd the equation of the tangent plane at the point ay sin9c7y7 77 lt272gt7 P Mir6 The Chain Rules a function ag2 where 57tystzst the partial derivatives are as follows 3f 7 3m 3g 32 E 7 Vf ltE757Egt7 etc Find 67106771 when 5 17t 27139 for the function 35 3t wxyyzxz xscostyssint7 zt The dimensions of a box are growing at the following rates length 3ftmin7 width 2ftmin7 height 05ftmin Find the rate the Volume is changing over time 4 w 2 w w woo mp w on w 00 The Gradient and the Chain Rule Let facy x7 4217 a Sketch the level curve mm b Calculate and sketch WW sketch the tangent line to the level curve at that point Use a vector c Fl and parametrization of the line d Find the equation of the tangent plane to facy at 21 Batman is trying to escape a burning building He must follow a path given by at 2 sint e2 0931 in meters t in minutes If the temperature in degrees centigrade at any point in the building is given by Tay 10at 7 y 50 h the temperature is changing over time when t 7r 6 minutes D0 not namalzztz the tangent watt r t Partial Derivatives amp Tangent Planes Without appealing to Clalraut s Thm nd all second partials for facy rig572033134431 The temperature at any point 0231 on a steel plate is given by Ta y 500709731 74ny kelvlns egg in feet What is the rate of change of temperature with distance at the point 23 in the direction of a the weeds and b the yyaxls Find the equation of the tangent plane at 71 1 72 Way r 7 3067213 Limits amp Partial Derivatives TF Along the yraxls 2 7 2 lim amp 0 Along the xeaxis lim i 0 when x 217 womb 06 27 Therefore lim 0 moan as y 2 lim f y 2 WWW x 21 Evaluate along the path y at and y 032 39 40 4 H 4 D 43 44 4 CT 4 47 48 03 Determine the region on which fxy ln4 7 2 7 12 is cts Calculate partial derivatives with respect to z and then with respect to y f 9671 may 2113 Way 7 95 1 WW 1MP 1 ky siny2 1 cosy Arc Length and Curvature Find an expression for the curvature of the curve given by Ft i tj 1915 When is the curvature the greatest What happens to the curvature as t 7 00 What does this mean Prove that the curvature of any line is zero Find the curvatures of circles of radius 2 and 12 Conjecture what the curvature of a circle of radius 7 would be Can you prove this Derivatives of Vector Functions For the curve given by Ft 1 2ti 1 t 7 t2j 17tt2 71517 a Find the vector equations of the tangent line to the curve at the point 111 b Find the unit tangent vector at that point Consider Ft t 7 sint1 7 costgt a Find F t at t 7T 7r2 Find the unit tangents at those points b Find all the places where the tangent vector is the zero vector What does this mean Vector Functions Describe Ft tsin7rtcos7rtgt as 0 g t S 1 ln particular7 what curve is described and how is it traversed TF For a given space curve7 there is exactly one way to pararnetrize that curve Find a direction vector for the line given by Ft 4 7 ti 2 5tj Recall that this is analogous to slope7 for lines in R2 Math 254 Practice Problems luau Nov 7 Justify your reasoning Iterated Integrals in Polar Coordinates J Fz sinrey dydm sinr d1 e dy 2 F If the temperamre on a plate is given by Tc y SOUm393 13 Kelvins m y in meters then at the point 1 0 in the direction 11 1 the Lemperati1re is changing an a rate of 60 Kelvinsmeter J 3 T For any function j39rry frcy dy 0 0 4 By changing no polar coordinates nd the volume under Lhe surface 1 y my over the interseclvion of Lhe regions 1 S 2 y2 5 4 and 0 S y 5 1 Sketch the region as 9 J 17 X gt1rawf9549 7 r1 31439 WI 353 W 31AXSWSQ XJQgt Wi3939 93 r V 39 wear4 a4 lsrsl ngriffi cvv lo LL 3 casma Maia Math 254 Practice Problems Wed Nov 16 Justify your reasoning Change of Variables Let R be the region bounded the lines a y 171 l 11 y 3r y 1 Using an appropriate change of variables evaluate 41 y sinz dA Don t 7 forget the Jacobirm 2 Using z psin 9quot cos H y psin n sin 6 2 mos 77 prove that the Jacobian for switch ing a triple integral to Spherical coordinates is In sin q 3 Using 1 7003 63 rain 6 z z39pr0v e that the Jacobian for snitchng a triple integral to Cylindrical coordinates is 39r 9 x V quot2 h 2 V z I 39I l l Io2 7 Um y 397 quotl x 3 l A X jJR7xj5MQ 3MAJ j M5va 41Mka w X j gm l lu V M 5M9 no 059 quot ayl wal smwm 35 P 39 P 9 Q 3 91M 5wst QJiWC J Q Gm 539 cOst 1mcpcoscp 939 expand Joan Sm 1 9 Ms W N3 coscp O J Q SM WIC 13 and sru QcOeQ 915W 05360 W99 56 3 9l n I39cgem39el 95W50 9 7Lst Casiwsnwy 3019 use V39Sin6 O 3 909 5M rune O rco516rsm76 fj o O l Math 254 Practice Problems Fri Oct 21 Justify your reasoning The Chain Rules For a function fx y where si ys i zs39 the partial derivatives are as follows 91 flt gtqem as 65 5 5quot as 1 Find when 5 131 27139 or the function 5 wzyyzzz 3530081 yzssinL zt 2 The dimensions of a box are growing at the following raLes length 3fimin Width 2ftmin I height Daftmin 41f 5 1 f 91 Find the mm the Volume is changing over time 9 X l 9 99 9 251 go 1J39x 9593 932 95 53quot lt34quot X213quotgt39 ltC Ytl SFA l O gZvo5f 4 x4 E5nquotf 4 a 4 O aw 2w 3 Gx9gtltgt lts swsws lgt Ta55w XE5 COS 6 Ljgtltm 0 4 A quot9quot zQ21i M x a VaZwk 34f m wm m AV Maggy 41de 39lt f 2w ah iiiWar 3 mkltkjwgt lt77 9 7 wko71h 1w 06 AAM Math 254 Practice Problems Mon Oct 10 Justify your reasoning Limits amp Partial Derivatives y 1 TF Along the y axis7 w0113300m 039 y lim 7 0 Along the x axis7 axHM 954 22 Therefore lim 4x7 maHMO z y 2 r y 2 Evaluate hm 7 alon the ath z and x2 wyH00 94 112 g p y y 3 Determine the region on which ay ln4 7 x2 7 12 is cts Contrary to what I said in class7 fmxy refers to the partial derivative with respect to x so that y is being held constant 4 Calculate partial derivatives with respect to z and then with respect to y f 9671 may 27513 WW 7 95 1 WW 111 1 May siny2 1 cosy Math 254 Practice Problems Fri Oct 14 Justify your reasoning Partial Derivatives amp Tangent Planes H Without appealing to Ciairautquots Thin nd all Second partials for fzy 33231 Qxyg 4y N i The temperature at any point 1 y 011 a steel plate is given by TL39 y 500 xzy 4y2m kelvins z y in feet What is the rate of change of temperature with distance at the point 273 in the directionb a the x aga39s and b the y axis Du e co Find the equation of the tangent plane at 1 1 2 fryx4 3z2y3 71 W 25 5 3 C0031 2X5 233 gjcty My 32 2 M33 MW ijySjltj gtltj 4 1 2 11043 gm ij Jng mm 40 943 2 4 6 KH Ta C 3 Ax jx 73023 0 223 52 Ft pm d mam 2 KH MM x outtaat LiSKf pramq 6 3quot 3 Unwind 3 740g 4X3 ijs 14H Lf z 3 1139 203 1x33 60 a note 3 DJ ms 90233 1 3 oz ozcx7j a2gt Q 7 ww Math 254 Practice Problems Mon Oct 24 Justify your reasoning Local amp Absolute Extrema 1 Find the critical points of f1y x3 4 y3 12mg and determine if they are maxima minima or saddle points For absolute extrema consider the boundary of a closed set in R2 Just like in Calc I we must also check what happens along the boundary 392 Find the local extrema using the methods above See if you can also check the bound aries of the region fxy sinny on the closed region 0 S r 3 77 O S y S 1 D 4 3833 0 3323912xquot 7W I 2 11115 gt 7X 3911 X0 3 l2 7 Q gt T 75X I L Xx x7l 0 Xx 7 ax 143 3910 6x 42 93 Gj 7 642 2541 6quot 3019 D lt 0 SAM C m I CNN Olamgt0 AKOHVO MM W 7 2 x I j oily O KCosQltj2 5MXD Xquot mo l0 003 M 171 7 I 31 5 M83 of 605le Ky 504035 91355quot 391 x 1 I T W 7 t x95M1 4 CDSCX39X into a 7 9 JR J j Dfo l 0 l Or SAddle D 40 3d 539MCXI whitRadar 39 Y 39 X7rs fmgsmryl LLch ar cmw39 w MM wa r 31 MJ was chm W mm mm L6 107 MM Max x Lquot J U 390 39 af my Math 254 Practice Problems Wed Nov 23 Justify your reasoning Vector Line Integrals l T If Iquot is a gradient vector field then the line integral of 13 along every curve is zero 2 F MO is the path from 0010 gt 100 r 110 000 then fa Vf 09F 0 3 Let F 113 1i 332112 1 and C be the upper semicircle going from 00 to 20 Option 1 Find a parametrization for C and setup the integral If F 17quot You should be able to come up with a parametrization for C in terms of sin cos Option 2 Verify that 14quot is a gradient vector eld Then nd a potential function f and apply the fundamental theorem Option 3 Can you think of a third way to solve this problem 0 1 quot Mg 92334 cmw is Zora lt2 VF 1395 x mmm m vecch Fred So u M 1 d an u endpoint 0 mW 39 iwa 3 fowl J 339 Wi k ltl Cosf sad 0 sts39lr 397 Fr f n t l I n sad Cost w cosama 70 if Fem Melt awful M9 we a g i ah 331 7 43 amenMllw be a gradual quot131 332 Vu r rwflmi j v3l dx 3Ix 39 9 wwi 3xfm 3x32lgt5399lgt5 w3K QUFMM L mm 39xyX33XIJK b A Maw Hm 2amp2 5mm 7 mwdr d we m 1an 106m 005 90 MIL w MITL 399 t I 39fgt lt00gt flt 7r0gt O t i Z39A QI7lt9ogtZQ 2 Math 254 Practice Problems Mon Nov 21 Justify your reasoning Scalar Line Integrals Find the mass of a Wire that goes from the origin to 11 along a straight line and then back to the origin along y 2 and with density function pa7y 7 kgm F2 1 t 1 if may be a useful parametrization IO Does the orientation of a curve the direction the curve is travei Sed with increasing t affect a scalar lineintegral You should think about the structure of the line integral Also compare your answer from 1 by using F2 L 12 0 W Git gawkf riggingng ll f 57 0 1CI39tXlH m 1 if M I li quot 05 904 let 7 nal 24 time 90 0007 z 5f 7k 115ml Ho t b a 1 NW 391 dH A M f wli tk le i39m x W 1K 5 i MSJ Ar Lll il F5 2M9 posiflwtawi ea W 3m y 4M am In atWNW 55 1 SW Ow Wizard is winged 10y 5 74 NiemfafanMA l l 7 M50 Ef l f V SD39tJHWaltsig inqmm Math 254 Practice Problems Fri Sept 30 Justify your reasoning Derivatives of Vector Functions 1 For the curve given by Ft 1 2ti 1 t 7 t2j 1 7 t 1amp2 7 t3ic a Find the vector equations of the tangent line to the curve at the point 111 b Find the unit tangent vector at that point P Consider Ft t 7 sin1t7 1 7 costgt a Find F t at t 7T7 7r2 Find the unit tangents at those points b Find all the places where the tangent vector is the zero vector What does this mean Final Exam Study Guide The nal exam will be cumulative Here is a rough outline of the material covered in this course Throughout look for connections Chapter 13 Vector Functions There Will be no questions directly from this chapter39 this served as background for all the interesting stuff we did later What is a vector function 77t What does Ft represent How does this relate to arclength What is a unit tangent vector Chapter 14 Partial Derivatives 0 Basic partial derivatives With respect to Ly 2 o The gradient vector of a function and its interpretations O Tangent planes to surfaces nding its equation 0 The Chain Ruies in multiple variables 0 Directional derivatives 0 Basic optimization problems nding local extrema on surfaces The second derivative test 0 Optimization With constraints and Lagrange Multipliers Chapter 15 Multiple Integrals 0 Basic double iterated integrals over rectangles then extended to regions bounded by cts curves Applications to mass etc 0 Converting to Polar Coordinates r dr d9 0 Triple iterated integrals Recall that the dif culty here is in setting up the region in space over Which to integrate and converting that into proper limits of integration 0 Converting to cylindrical and spherical coordinates r dr d9 d2 p2 sinqb dpdq d 0 General change of variables Transformations of regions in the plane the Jacobian This is useful in 2 cases sometimes simultaneously a changing variables to make the integrand easier or b changing to a different coordinate system r 939 p 45 939 u v to make the region easier Chapter 16 Vector Fields and Line Integrals O A vector eld is a function Which assigns a vector to every point in its domain Be able to identify vector elds from their pictures or equations 0 Scalar line integrals how much area under a surface along a curve think of one side of a fence or curtain f0 ds 0 Vector Line Integrals How much does the vector eld point in the same direction tangential as our curve C F d O Conservative vector elds eg Vf F Test for conservative Finding potential functions O The Fundamental Thin of Vector Line lntegrals Independence of path in conservative vector elds If C is given t then O War fjv m we dt mm 7 ea Chapter 17 Just Divergence and Curl O div F V Measures expansion of the eld at a point SCALARl O curl F V X Is a VECTOR that gives the axis and speed of induced spin in a eld at a point You can ll an 8l5gtlt11 sheet one side With Whatever you want on it to reference during the nal exarnl Your best study guides Will be the previous exams practice problems quizzes and homework Math 254 Practice Problems Mon Oct 31 Justify your reasoning Iterated Integrals 1 Calculate 1 2y2 dydz where R 0 2 X 04 R 1 4 gt4 1 2 Verify Lhat mag3 dyda Igya dutdy o 2 2 o 2 y 9 quQj5lJJXJ fw I jg ix j Ajix o 2 x r qux Hg1x 98 393ng Yd L l R 53931quot 3 ax ii quotQj 5M 1 a j J I lxg 31x Q3lj 533 479 3 1X19L1 39 Ll Eli 5y 3amp9 15 Lt bOxl 33960X2dx 393 34431 939 Math 254 Practice Problems Wed Nov 2 Justify your reasoning Iterated Integrals over General Regions 1 Calculate the volume under the surface f zy 23 over the region bounded by z 2 4 y2 in the rst quadrant Sketch the region 2 The temperature on a triangular plate is given by Tz y 1206quot2 degrees celsius Find the average temperature on the plate if it has vertices 32003 0 m 1x3 M 111 1 3 5 tilt 610 extM 1 A l 1 3 2001 63 A j Adm 54 RAM 393 3t 7ULS 391st H 22 322Wquot V a 0 2 f j ae g lj mod 5 o 3 J1 J23 o jiaix quot0553 W 05 L0Llt 99 Math 254 Practice Problems Wed Sept 28 Justify your reasoning Vector Functions 1 Describe 770 t sin7nt7 cos7rtgt as 0 g t g 1 ln particular7 what curve is described7 how is it oriented7 and how is it traversed 2 TF For a given space curve7 there is exactly one way to parametrize that curve 3 Find a direction vector for the line given by 770 4 7 ti 2 5tj Recall that this is analogous to slope7 for lines in R2 Math 254 Practice Problems Wed Oct 5 Justify your reasoning ArcLength and Curvature H Find an expression for the curvature of the curve given by 770 i tj tzk When is the curvature the greatest What happens to the curvature as t 00 What does this mean E0 Prove that the curvature of any line is zero 9 Find the curvatures of circles of radius 2 and 12 Conjecture what the curvature of a circle of radius 7 would be Can you prove this Math 254 Practice Problems Fri Oct 26 Justify your reasoning Lagrange Multipliers What is the largest open top box that can be made with 12m2 of material Vl w h lwh 12 21h 2wh 111 By setting VV AVS Where in this example we think of Sl w h as the surfacearea function at the level curve 3 12 This gives us the system 0 wh M217 in lh 2 L l lw 2Al 10 12 2th th 1211 2 Lagrange Multipliers give us a powerful tool for nding absolute extrema in the case when the boundary is not a rectangle First set the grad equal to zero to nd all local extrema inside the region Then use Lagrange multipliers to nd the maximum and minimum values of the function where the boundary is your constraint f y 2x2 By2 4m 5 on the closed disk 302 y2 g 16 Here we can think of the constraint as gz y 32 y2 at the level curve 2 16 SIMCC 4w 1 70 001 12hm3 7 O Qk I M aw wk 1 QMll O A1w Qhwz l GA A RMAELU N h f 2 akyquot 2610 JkX2lA A rmil M M 12 gt k mm W I 2651 4 k3 Q VIO 5 x 4217 I crffafdp 01 2 j 601 quot4quot 10 M x Cmt d 7 39duy39j0 WA 6 A23 7 av 0 Exam Cat 1 0 W x19 a 03 quj x 10 lsEN 1024 lbzqwjg uziwa 353 3 MC W 1 if 5 g4 45 izr 7lt mx Qx 3 Math 254 Practice Problems Fri Nov 4 Justify your reasoning Triple Integrals 1 Find the mass of the solid bounded by the surface 2 9 x2 y2 in the rst oatamt cyz 2 O with density function pr1T 312 3212 Set up the integral with at least 3 different orders of integration 2 1xa a x4 50 x3394 391 E 4 xa j 395 Jm i39 j 50 So i 355 dellleX 1 30 Z1q X2 11 0st B w o 92 e 7 X2 0 2x53 qu zi gi d3 dzdx 3 E 2 N1 32 gtltOV 241 3 Csxem L 1 1 3 E jig 1339 dxdlzdj 701 W55 72

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