15 MULTIPLE INTEGRALS

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1 5 MULTIPLE INTEGRALS 5. Double Integrals over Rectangles. (a) The subrectangles are shown in the figure. Thesurfaceisthegraphof ( ) and 4,soweestimate ( ) (b) () +(4) +(4) +(44) +(6) +(64) 4(4)+8(4)+8(4)+6(4)+(4)+4(4) 88 () +() +() +() +(5) +(5) (4)+(4)+(4)+9(4)+5(4)+5(4) 44. (a) The subrectangles are shown in the figure. Here andweestimate (b) ( ) +() +() +(4 ) +(4) +(4) ()()+()+()()+()()+()+()() () +() +() +() +() +() ()+()+()+()+()()+(7)() 8. (a) Thesubrectanglesareshowninthefigure. Since,weestimate (b) +() + +() ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. 5

2 5 CHAPTER 5 MULTIPLE INTEGRALS 4. (a) The subrectangles are shown in the figure. Thesurfaceisthegraphof ( ) + + and 4, so we estimate (+ +) () (b) (+ +) ( ) (a) Each subrectangle and its midpoint are shown in the figure. Theareaofeachsubrectangleis,soweevaluate at each midpoint and estimate ( ) (5) +(5) + (5) +(5) ()+()()+()+() 4 (b) The subrectangles are shown in the figure. In each subrectangle, the sample point closest to the origin isthelowerleftcorner,andtheareaofeachsubrectangleis. Thus we estimate ( ) 4 4 () +(5) +() +(5) + () +(5) +() +(5) + () +(5) +() +(5) + () +(5) +() +(5) +(5) +(6) +(4) +() +() +() +() + + +() c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

3 SECTION 5. DOUBLE INTEGRALS OVER RECTANGLES 5 6. Toapproximatethevolume,let betheplanarregioncorrespondingtothesurfaceofthe waterinthepool,andplace oncoordinateaxessothat and correspondtothe dimensionsgiven. Thenwedefine ( )tobethedepthofthewaterat( ),sothe volumeofwaterinthepoolisthevolumeofthesolidthatliesabovetherectangle [] []andbelowthegraphof ( ). Wecanestimatethisvolumeusing themidpointrulewith and,so. Eachsubrectanglewithits midpointisshowninthefigure. Then [(55)+(55)+(55)+(55)+(55)+(55)] ( ) 6 Thus,weestimatethatthepoolcontains6cubicfeetofwater. Alternatively,wecanapproximatethevolumewithaRiemannsumwhere 4, 6andthesamplepointsaretakento be,forexample,theupperrightcornerofeachsubrectangle. Then 5and 4 6 ( ) 5[ ] 5(4) 5 Soweestimatethatthepoolcontains5ft ofwater. 7. Thevaluesof ( ) 5 getsmalleraswemovefartherfromtheorigin,soonanyofthesubrectanglesinthe problem,thefunctionwillhaveitslargestvalueatthelowerleftcornerofthesubrectangleanditssmallestvalueattheupper rightcorner,andanyothervaluewillliebetweenthesetwo. Sousingthesesubrectangleswehave. (Notethatthis istruenomatterhow isdividedintosubrectangles.) 8. Divide into4equalrectangles(squares)andidentifythemidpoint of each subrectangle as shown in the figure. Theareaofeachsubrectangleis,sousingthecontourmaptoestimatethefunctionvaluesateachmidpoint,wehave ( ) ()()+()()+()()+(5)() Youcouldimprovetheestimatebyincreasing and tousealargernumberofsmallersubrectangles. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

4 54 CHAPTER 5 MULTIPLE INTEGRALS 9. (a) With,wehave 4. Usingthecontourmaptoestimatethevalueof atthecenterofeachsubrectangle, we have ( ) (b) ave ( ) (48) 55 () 6 [()+()+()+()] 4( ) 48. AsinExample4,weplacetheoriginatthesouthwestcornerofthestate. Then [88] [76](inmiles)isthe rectanglecorrespondingtocoloradoandwedefine ( )tobethetemperatureatthelocation( ). Theaverage temperature is given by ave ( ) ( ) () TousetheMidpointRulewith 4,wedivide into6regionsofequalsize,asshowninthefigure,withthecenter of each subrectangle indicated. Theareaofeachsubrectangleis ,sousingthecontourmaptoestimatethefunctionvaluesateach midpoint, we have ( ) 4 Therefore, ave approximately78 F. 4 [ ] 669(65) 78,sotheaveragetemperatureinColoradoat4:PMonFebruary6,7,was c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

5 SECTION 5. DOUBLE INTEGRALS OVER RECTANGLES 55.,sowecaninterprettheintegralasthevolumeofthesolid thatliesbelowtheplane andabovethe rectangle[] [6]. isarectangularsolid,thus for 5,sowecaninterprettheintegralasthevolumeofthesolid thatliesbelowtheplane 5 andabovetherectangle[5] []. isa triangularcylinderwhosevolumeis(areaoftriangle) Thus (5 ) 75. ( ) 4 for. Thustheintegralrepresentsthevolumeofthat partoftherectangularsolid[] [] [4]whichliesbelowtheplane 4. So (4 ) ()()()+ ()()() 4. Here 9,so + 9,. Thustheintegralrepresentsthevolumeof thetophalfofthepartofthecircularcylinder + 9thatliesabovetherectangle [4] []. 5. To calculate the estimates using a programmable calculator, we can use an algorithm similartothatofexercise4..9[et5..9]. InMaple,wecandefinethefunction ( ) + (callingitf),loadthestudentpackage,andthenusethe command middlesum(middlesum(f,x..,m), y..,m); togettheestimatewith squaresofequalsize. Mathematicahasnospecial Riemannsumcommand,butwecandefinefandthenusenestedSum commandsto calculate the estimates. estimate estimate estimate c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

6 56 CHAPTER 5 MULTIPLE INTEGRALS 7. Ifwedivide into subrectangles, But alwaysand points, lim foranychoiceofsamplepoints. areaof ( )( ). Thus,nomatterhowwechoosethesample ( )( )andso lim lim ( )( ) ( )( ). 8. Becausesin isanincreasingfunctionfor 4,wehavesin sin sin 4 sin. Similarly,cos isadecreasingfunctionfor,so cos cos cos. Thuson, 4 4 sin cos. Property(9)gives sin cos have sin cos 4 4.,sobyExercise7we 5. Iterated Integrals (5) 4() 5, () 4 () ( + ) (+) 5+ 5, ( + ) + + (+ ) + 4 (6 ) 4 4 ( 4) 4 4 (56 ) (4 ) (4 9 ) 4 (8 4 ) (4 ) (4 ) 4 7 ( 7) ( ) [asinexample5] (4 )(64 ) ( 4 ) 6. 5 cos 5 cos [byequation5] (+ cos ) 5 sin [5 ()](sin sin ) 6( ) sin c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

7 ln [ln ] + 5 ln [asinexample5] (ln ) 5 [substitute ln () ] (ln ) [(ln5) ] (ln)(ln5) 4 ln + 4 8ln+ ln4 ln 5 ln+ln4 ln SECTION 5. ITERATED INTEGRALS 57 ln+ ln+ ln [ ] ( ) or (6 +) (+ ) 4 5 (+ ) 5 (+ ) 5 (+ ) (+ ) (+ ) 6 [substitute + inthefirstterm]. 6 ( 6 ) ( ) (6 ) 6 + ( + ) [( +) ] [( +) 4 ] 5 ( +) sin sin [asinexample5] + sin [( ) ( )] ( cos) ( ) sin sin (+) [(+) ] 5 (+) [(5 ) ( )] or sin( ) sin( ) [cos( )] cos( ) cos sin( ) sin sin sin sin( ) sin 7. (+ ) + ( ) (+ ) + + (4 ) ( ) 4 + (ln ln) (7+7) 9ln + ln( +) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

8 58 CHAPTER 5 MULTIPLE INTEGRALS sin(+) cos(+) sin sin + 6 (+ ) cos cos + sin sin cos +cos tan [byintegratingbypartsseparatelyforeachterm] ln(+) (+)ln(+) (ln ) (ln ) ln ln(+) ln(+) ln [by integrating by parts] ( +) [ln(++)] [ln(+) ln(+)] (+)ln(+) (+) (+)ln(+) (+) [by integrating by parts separately for each term] (6ln6 65ln5+5) (4ln4 4ln+) 6ln6 5ln5 4ln4+ln. ( ) 4 for and. Sothesolid istheregioninthefirstoctantwhichliesbelowtheplane 4 andabove[] []. 4. for and. Sothesolidisthe region in the first octant which lies below the circular paraboloid andabove[] []. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

9 SECTION 5. ITERATED INTEGRALS Thesolidliesundertheplane or ++ 5 so (+ + 5 ) (++ 5 ) ++ 5 (9+6) ( ) (5 +9) () 5 6. ( +) ( +) + ( ) ( ) (+ sin ) cos (+ + ) (+ ) Hereweneedthevolumeofthesolidlyingunderthesurface sec andabovetherectangle [] [ 4]in the -plane. 4 sec 4 sec 4 tan ( )(tan tan) ( ) 4. Thecylinderintersectsthe -planealongtheline 4,sointhefirstoctant,thesolidliesbelowthesurface 6 andabovetherectangle [4] [5]inthe -plane. 5 4 (6 ) 4 (6 ) (64 64 )(5 ) 64. Thesolidliesbelowthesurface + +( ) andabovetheplane for, 4. Thevolume ofthesolidisthedifferenceinvolumesbetweenthesolidthatliesunder + +( ) overtherectangle [] [4]andthesolidthatliesunder over [+ +( ) ] 4 () 4 (+ +( ) ) ( ( ) ) [] []4 4 +( ) [ ()][4 ] 4 + ( ) ( ) 4 ()(4). Thesolidliesbelowtheplane + andabovethesurface for, 4. Thevolumeofthesolidis + the difference in volumes between the solid that lies under +overtherectangle [] [4]andthesolidthat liesunder over. + [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

10 5 CHAPTER 5 MULTIPLE INTEGRALS 4 (+) 4 [(4+6) (+)] ln (ln5 ln)(8 ) (8+ ) 8ln5 4 8ln5. InMaple,wecancalculatetheintegralbydefiningtheintegrandasf and then using the command int(int(f,x..),y..);. In Mathematica, we can use the command Integrate[f,{x,,},{y,,}] Wefindthat Wecanuseplotd (in Maple) or PlotD(in Mathematica) to graph the function. 4. In Maple, we can calculate the integral by defining f:exp(-xˆ)*cos(xˆ+yˆ); and g:-xˆ-yˆ; andthen[since cos( + )for, ]usingthecommand evalf(int(int(g-f,x-..),y-..),5);. The 5 indicates that we want only five significant digits; this speeds up the calculation considerably. + In Mathematica, we can use the command NIntegrate[g-f,{x,-,},{y,-,}]. We find that ( ) cos( + ) 7. Wecanusetheplotdcommand(inMaple)orPlotD (in Mathematica) to graph both functions on the same screen. 5. is the rectangle [] [5]. Thus, () 5 and ave () ( ) () 4 4,so 7. function so ave () ( ) [(+) (+) ] 6 5 (+)5 5 (+) (+ ) 6 5 [(4+) ] 5 [(4+) ] [byequation5] but () 4 + isanodd 4 () by(6)insection4.5[et(7)insection5.5]. Thus. +4 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

11 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS 5 8. (+ sin + sin ) + sin + sin ()+ sin + sin ()()+ sin + sin But sin isanoddfunction,so sin sin by(6)insection4.5[et(7)insection5.5]and (+ sin + sin ) Let ( ) (+). ThenaCASgives ( ) and ( ). ToexplaintheseemingviolationofFubini stheorem,notethat hasaninfinitediscontinuityat()andthusdoesnot satisfy the conditions of Fubini s Theorem. In fact, both iterated integrals involve improper integrals which diverge at their lower limits of integration. 4. (a) Loosely speaking, Fubini s Theorem says that the order of integration of a function of two variables does not affect the value of the double integral, while Clairaut s Theorem says that the order of differentiation of such a function does not affect the value of the second-order derivative. Also, both theorems require continuity(though Fubini s allows a finite number of smooth curves to contain discontinuities). (b) Tofind,wefirsthold constantandusethesingle-variablefundamentaltheoremofcalculus,part: ( ) ( ) ( ). NowweusetheFundamentalTheoremagain: ( ) ( ). Tofind,wefirstuseFubini stheoremtofindthat ( ) FundamentalTheoremtwice,asabove,toget ( ). So ( ). ( ),andthenusethe 5. Double Integrals over General Regions [( ) ] (64 ). ( ) () () ()+ () ( ) (+) + + ( ) ( 4 ) (4 ) 4 (4 ) 6 4 cos( ) cos( ) cos( ) sin( ) (sin sin) sin c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

12 5 CHAPTER 5 MULTIPLE INTEGRALS (+ ) (+) (+) (+) 4 7. [ ( )] ( + ) sin 5 + (ln ln) ln 5 + []sin sin cos +sin cos +sin +sin ln ln ln 4 4 ln ln integrate by parts with sin integrate by parts with ln. (a) Attherightwesketchanexampleofaregion thatcanbedescribedaslying betweenthegraphsoftwocontinuousfunctionsof (atypeiregion)butnotas lyingbetweengraphsoftwocontinuousfunctionsof (atypeiiregion). The regionsshowninfigures6and8inthetextareadditionalexamples. (b) Nowwesketchanexampleofaregion thatcanbedescribedaslyingbetween thegraphsoftwocontinuousfunctionsof butnotaslyingbetweengraphsoftwo continuousfunctionsof. ThefirstregionshowninFigure7isanotherexample.. (a) Attherightwesketchanexampleofaregion thatcanbedescribedaslying betweenthegraphsoftwocontinuousfunctionsof (atypeiregion)andalsoas lyingbetweengraphsoftwocontinuousfunctionsof (atypeiiregion). For additionalexamplesseefigures9,,,and4 6inthetext. (b) Nowwesketchanexampleofaregion thatcan tbedescribedaslyingbetween thegraphsoftwocontinuousfunctionsof orbetweengraphsoftwocontinuous functionsof. TheregionshowninFigure8isanotherexample. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

13 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS 5. AsatypeIregion, liesbetweenthelowerboundary andtheupper boundary for,so {( ), }. Ifwe describe asatypeiiregion, liesbetweentheleftboundary andthe rightboundary for,so {( ), }. Thus ( ) or ( ). 4. Thecurves and intersectatpoints(),(9). AsatypeIregion, isenclosedbythelowerboundary andtheupperboundary for,so ( ),. Ifwedescribe asa typeiiregion, isenclosedbytheleftboundary andtherightboundary for 9,so ( ) 9,. Thus (9 4 ) (9 5 ) or Thecurves or +and intersectwhen + ( )(+),,sothepointsof intersectionare( )and(4). Ifwedescribe asatypeiregion,theupper boundarycurveis butthelowerboundarycurveconsistsoftwoparts, for and for 4. Thus {( ), } {( ) 4, }and + 4. Ifwedescribe asatypeiiregion, isenclosedbytheleftboundary andtherightboundary +for,so ( ), + and +. Ineithercase,theresultingiteratedintegralsarenotdifficulttoevaluatebuttheregion is c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

14 54 CHAPTER 5 MULTIPLE INTEGRALS moresimplydescribedasatypeiiregion,givingoneiteratedintegralratherthanasumoftwo,soweevaluatethelatter integral: + + (+ ) ( + ) AsatypeIregion, {( ) 4, 4}and AsatypeIIregion, {( ) 4, }and 4. Evaluating requiresintegrationbypartswhereas doesnot,so theiteratedintegralcorrespondingto asatypeiiregionappearseasiertoevaluate cos sin sin cos ( cos) 8. ( +) ( +) + ( ) [(7 ) ( )] (8 4 ) ( ) ( 4 ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

15 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS ( ) [Or,notethat4 4 isanoddfunction,so 4 4.]. [( ) () ] ( 6 +9) ( +) + ( ) ( )+( ) ( ) ( )+( ) (+ ) + 4 ( ) ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

16 56 CHAPTER 5 MULTIPLE INTEGRALS 6. ( + ) + ( + ) (6 ) 6 6( ) ( ) ( ) ( ) (4 4 ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

17 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS 57.. Bysymmetry,thedesiredvolume is8timesthevolume inthefirstoctant. Now ( ) Thus 6.. Fromthegraph,itappearsthatthetwocurvesintersectat and at. Thusthedesiredintegralis 4 4 ( 5 ) Thedesiredsolidisshowninthefirstgraph. Fromthesecondgraph,weestimatethat cos intersects at 79. Thereforethevolumeofthesolidis 79 cos 79 cos 79 cos 79 (cos ) cos +sin 79 4 Note: There is a different solid which can also be construed to satisfy the conditions stated in the exercise. This is the solid boundedbyallofthegivensurfaces,aswellastheplane. Incaseyoucalculatedthevolumeofthissolidandwantto checkyourwork,itsvolumeis 79 + cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

18 58 CHAPTER 5 MULTIPLE INTEGRALS 5. Thetwoboundingcurves and intersectat(±)with on[]. Withinthis region,theplane + +isabovetheplane,so (++) ( ) (++ ( )) (++8) + +8 ( )+ ( ) +8( ) ( ) ( ) 8( ) ( ) Thetwoplanesintersectintheline,,sotheregionof integrationistheplaneregionenclosedbytheparabola andthe line. Wehave+ for,sothesolidregionis boundedaboveby +andboundedbelowby. (+) () (+ ) ( + 4 ) ( ) 7. Thesolidliesbelowtheplane or ++ andabovetheregion {( ) } inthe -plane. Thesolidisatetrahedron. 8. Thesolidliesbelowtheplane and above the region ( ) inthe -plane. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

19 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS Thetwoboundingcurves and + intersectattheoriginandat,with + on(). UsingaCAS,wefindthatthevolumeis + + ( 4 + ),984,75,66 4,549,55 4. For and, + 8. Also,thecylinderisdescribedbytheinequalities,. Sothevolumeisgivenby (8 ) ( + ) [using a CAS] 4. Thetwosurfacesintersectinthecircle +, andtheregionofintegrationisthedisk : +. UsingaCAS,thevolumeis ( ) ( ). 4. Theprojectionontothe -planeoftheintersectionofthetwosurfacesisthecircle + + +( ),sotheregionofintegrationisgivenby, +. Inthisregion, + so,usingacas,thevolumeis + [ ( + )] 4. Because the region of integration is {( ) } {( ) } wehave ( ) ( ) ( ). 44. Because the region of integration is wehave ( ) 4 ( ) 4 4 ( ) ( ) 4 ( ). 45. Because the region of integration is we have {( ) cos } ( ) cos cos ( ) ( ) cos ( ). c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

20 5 CHAPTER 5 MULTIPLE INTEGRALS 46. Because the region of integration is ( ) 4 ( ) 4 4 we have 4 ( ) ( ) 4 4 ( ). 47. Because the region of integration is {( ) ln, } {( ), ln} we have ln ( ) 48. Because the region of integration is we have 4 arctan ( ) ln ( ) arctan, 4 ( ) tan, 4 ( ) ( ) 4 tan ( ) ( ) cos( ) cos( ) cos( ) sin( ) (sin sin) cos( ) ln + (ln9 ln) ln9 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

21 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS 5 5. ( ) ( ) ( ) 5. arcsin cos +cos sin cos +cos cos +cos sin cos +cos sin Let cos, sin, (sin ) (6 ) {( ), + } {( ), + } {( ), } {( ), }, alltypei [bysymmetryoftheregionsandbecause ( ) ] ( ), ( ),,bothtypeii. + ( 4 + ) + ( 4 + ) ( )+( 7 ) 5 + c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

22 5 CHAPTER 5 MULTIPLE INTEGRALS 57. Here ( ) +,and ( + ) ( + ) so 6 ( + ) since isanincreasingfunction. Wehave () 4 6,sobyProperty, 6 () ( + ) () 6 6 ( + ) 6 orwecansay 844 ( + ) 964. (Wehaveroundedthelowerbounddownandtheupperbounduptopreservethe inequalities.) 58. isthetrianglewithvertices(),(),and()so () ()(). Wehave sin4 (+) forall,, andpropertygives () sin4 (+) () sin4 (+). 59. Theaveragevalueofafunction oftwovariablesdefinedonarectangle was definedinsection5.as ave ( ). Extendingthisdefinition () togeneralregions,wehave ave ( ). () Here {( ) },so () ()() and ave ( ) () Here ( ),so () and ave ( ) () cos sin cos( ) sin( ) sin ( sin) 6. Since ( ), ( ) by(8) ( ) by(7) () ( ) ()by(). 6. ( ) ( ) + ( ) ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

23 SECTION 5. DOUBLE INTEGRALS OVER GENERAL REGIONS 5 6. Firstwecanwrite (+) +. But ( ) is anoddfunctionwithrespectto [thatis, ( ) ( )]and is symmetricwithrespectto. Consequently,thevolumeabove andbelowthe graphof isthesameasthevolumebelow andabovethegraphof,so. Also, () () 9since isahalf diskofradius. Thus (+) Thegraphof ( ) isthetophalfofthesphere + +,centeredattheoriginwithradius,and isthediskinthe -planealsocenteredattheoriginwithradius. Thus representsthe volumeofahalfballofradius whichis Wecanwrite (+) +. representsthevolumeofthesolidlyingunderthe plane andabovetherectangle. Thissolidregionisatriangularcylinderwithlength andwhosecross-sectionisa trianglewithwidth andheight. (Seethefirstfigure.) Thusitsvolumeis. Similarly, representsthevolumeofatriangularcylinderwithlength, triangularcross-sectionwithwidth andheight,andvolume. (Seethesecondfigure.) Thus (+) Inthefirstquadrant, and arepositiveandtheboundaryof is +. But is symmetricwithrespecttobothaxesbecauseoftheabsolutevalues,sotheregionof integration is the square shown at the left. To evaluate the double integral, we first write (+ sin ) + sin. Now ( ) isoddwithrespectto [thatis, ( ) ( )] and issymmetricwithrespectto,so. Similarly, ( ) sin isoddwithrespectto [since ( ) ( )]and issymmetricwithrespectto, so sin. isasquarewithsidelength,so () 4,and (+ sin ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

24 54 CHAPTER 5 MULTIPLE INTEGRALS Now isoddwithrespect to and isoddwithrespectto,andtheregionofintegrationissymmetricwithrespecttoboth and, so. representsthevolumeofthesolidregionunderthe graphof andabovetherectangle,namelyahalfcircular cylinderwithradius andlength(seethefigure)whosevolumeis (). Thus To find the equations of the boundary curves, we require that the -valuesofthetwosurfacesbethesame. InMaple,weusethecommand solve(4-xˆ-yˆ-x-y,y); and in Mathematica, we use Solve[4-xˆ-yˆ-x-y,y]. We find that the curves have equations ± Tofindthetwopointsofintersection ofthesecurves,weusethecastosolve+4 4,findingthat ± 4. So,usingtheCAStoevaluatetheintegral,thevolumeofintersectionis (+ 4) + ( 4) [(4 ) ( )] Double Integrals in Polar Coordinates. Theregion ismoreeasilydescribedbypolarcoordinates: ( ) 4, Thus ( ) 4 (cos sin ).. Theregion ismoreeasilydescribedbyrectangularcoordinates: ( ),. Thus ( ) ( ).. Theregion ismoreeasilydescribedbyrectangularcoordinates: ( ), +. Thus ( ) (+) ( ). 4. Theregion ismoreeasilydescribedbypolarcoordinates: ( ) 6,. Thus ( ) 6 (cos sin ).. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

25 5. Theintegral 4 4 representstheareaoftheregion {( ), 4 4},thetopquarterportionofa ring(annulus) SECTION 5.4 DOUBLE INTEGRALS IN POLAR COORDINATES 55 (4 ) 4 6. Theintegral sin representstheareaoftheregion {( ) sin, }. Since sin sin + +( ), istheportioninthesecondquadrantofadiskof radiuswithcenter(). sin sin sin ( cos) sin + 7. Thehalfdisk canbedescribedinpolarcoordinatesas {( ) 5, }. Then 5 (cos ) (sin ) cos sin 5 4 cos ( ) Theregion is ofadisk,asshowninthefigure,andcanbedescribedby {( ), 4 }. Thus 8 9. ( ) (cos sin ) 4 (cos sin ) 4. sin +cos 4 (+ ) sin( + ) sin( ) sin( ) + cos( ) (cos9 cos) (cos cos9) 4 (sin ) ( cos) ( ) ( ) sin sin c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

26 56 CHAPTER 5 MULTIPLE INTEGRALS. ( 4 ) ( 4 ). cos + cos cos. Forthesecondintegral,integratebypartswith, cos. Then cos + [sin +cos ] (sin+cos ).. istheregionshowninthefigure,andcanbedescribed 4. by {( ) 4 }. Thus arctan() 4 arctan(tan ) since tan. Also,arctan(tan ) for 4,sotheintegralbecomes Oneloopisgivenbytheregion + 4, ( ) + cos cos {( ) 6 6, cos},sotheareais 6 cos 6 cos cos sin6 6 cos (8cos ) (8cos4 ) 8 8 cos sin + (+sin cos ) cos6 6. Bysymmetry,theareaoftheregionis4timestheareaoftheregion inthefirstquadrantenclosedbythecardiod cos (seethefigure). Here {( ) cos },sothetotalareais 4() 4 4 cos 4 cos ( cos ) cos + (+cos) sin sin ( cos +cos ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

27 SECTION 5.4 DOUBLE INTEGRALS IN POLAR COORDINATES Inpolarcoordinatesthecircle( ) + + is cos cos, andthecircle + is. Thecurvesintersectinthefirstquadrantwhen cos cos,sotheportionoftheregioninthefirstquadrantisgivenby {( ) cos }. Bysymmetry,thetotalarea istwicetheareaof : () cos cos 4cos 4 (+cos) (+cos) [+sin] + 8. TheregionliesbetweenthetwopolarcurvesinquadrantsIandIV,butin quadrantsiiandiiitheregionisenclosedbythecardiod. Inthefirst quadrant,+cos cos whencos,sothearea oftheregioninsidethecardiodandoutsidethecircleis +cos cos (+cos 8cos ) +cos cos +sin sin 4 + Theareaoftheregioninthesecondquadrantis +sin sin + 4 +cos +cos (+cos +cos ) +sin + + sin Bysymmetry,thetotalareais ( + ) Theparaboloid 8 intersectsthe -planeinthecircle + 9,so () c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

28 58 CHAPTER 5 MULTIPLE INTEGRALS. Thehyperboloidoftwosheets + intersectstheplane when +4 or +.Sothe solidregionliesabovethesurface + + andbelowtheplane for +,anditsvolumeis (+ ) 4. Thesphere + + 6intersectsthe -planeinthecircle + 6,so By symmetry, 6 [bysymmetry] 4 (6 ) ()( ) ( ) () (6 ) 4. Theparaboloid + + intersectstheplane 7when7 + + or + andwearerestricted tothefirstoctant,so (+ ) Thecone + intersectsthesphere + + when or +. So + + ( ) 6. Thetwoparaboloidsintersectwhen + 4 or +. So + [(4 ) ( + )] (4 4 ) 4 4( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

29 SECTION 5.4 DOUBLE INTEGRALS IN POLAR COORDINATES Thegivensolidistheregioninsidethecylinder + 4betweenthesurfaces and So ( 6 ) (6 ) 8. (a) Heretheregioninthe -planeistheannularregion + andthedesiredvolumeistwicethatabovethe -plane. Hence 4 + ( ) 4 ( ) (b) Across-sectionalcutisshowninthefigure. So + or 4. Thusthevolumeintermsof is sin( + ) sin sin [] cos (cos9 ) ( cos9). (cos ) (sin ) 4 cos sin cos sin 4 cos 5 5 cos cos (cos + sin ) 4 (cos +sin ) [sin cos ] 4 + c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

30 54 CHAPTER 5 MULTIPLE INTEGRALS. cos cos 8 cos 8 8 ( sin )cos sin sin 6 9. {( ), }, so ( + ) ( ) 4 4. Usingacalculator,weestimate {( ), },so + + (cos )(sin ) + sin cos + sin Thesurfaceofthewaterinthepoolisacirculardisk withradiusft. Ifweplace oncoordinateaxeswiththeoriginat thecenterof anddefine ( )tobethedepthofthewaterat( ),thenthevolumeofwaterinthepoolisthevolumeof thesolidthatliesabove ( ) + 4 andbelowthegraphof ( ). Wecanassociatenorthwiththe positive -direction,sowearegiventhatthedepthisconstantinthe -directionandthedepthincreaseslinearlyinthe -directionfrom ( ) to () 7. Thetraceinthe -planeisalinesegmentfrom( )to(7). Theslopeofthislineis 7,soanequationofthelineis 7 ( ) + 9. Since ( )is () independentof, ( ) + 9. Thusthevolumeisgivenby ( ),whichismostconvenientlyevaluated 8 usingpolarcoordinates. Then {( ), }andsubstituting cos, sin theintegral becomes sin sin sin +9 Thusthepoolcontains8 5655ft ofwater. cos (a) If,thetotalamountofwatersuppliedeachhourtotheregionwithin feetofthesprinkleris [ ++] ( )ft (b) Theaverageamountofwaterperhourpersquarefootsuppliedtotheregionwithin feetofthesprinkleris areaofregion ft (perhourpersquarefoot). Seethedefinitionoftheaveragevalueofa function on page [ET 979]. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

31 SECTION 5.4 DOUBLE INTEGRALS IN POLAR COORDINATES AsinExercise5..59, ave ( ). Here {( ) }, () so () ( )and ave () + ( ) ( ) ( ) ( ) ()( ) ( ) (+)( ) + 8. Thedistancefromapoint( )totheoriginis ( ) +,sotheaveragedistancefrompointsin totheoriginis ave + () [] cos sin sin cos sin 4 4 cos sin (a) ( + ) foreach. Then lim since as. Hence ( + ). (b) ( + ) foreach. Then,from(a), R ( + ),so lim ( + ) lim Toevaluate lim,weareusingthefactthattheseintegralsarebounded. Thisistruesince on[], whileon( ), andon( ),. Hence + + ( +). (c) Since and canbereplacedby, impliesthat ±. But forall,so. (d) Letting,,sothat or. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

32 54 CHAPTER 5 MULTIPLE INTEGRALS 4. (a) Weintegratebypartswith and. Then and,so lim lim + lim + + [byl Hospital srule] 4 [since isanevenfunction] 4 [byexercise4(c)] (b) Let. Then lim lim 4 [bypart(a)]. 5.5 Applications of Double Integrals. ( ) 5 5 (+4) (+5 4 8) 5 (6+4) C. ( ) + [] C. ( ) 4 4 [] 4 ()() 4, 4 ( ) (4)(), ( ) [] 4 4 () Hence 4,( ) ( ) (+ + ) (+ + ), ( ) (+ + ) (6+ + ),and ( ) (+ + ) (6+ + ). Hence, ( ) (6+ + ) (+ + ) (6+ + ) 4(+ + ) (6+ + ) 4(+ + ) (6+ + ) (+ + ). c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

33 SECTION 5.5 APPLICATIONS OF DOUBLE INTEGRALS (+) + + ( ) , 8 ( + ) , 8 (+ ) Hence 6,( ) Here {( ) 6 }. 6 (6 ) 6 4 4, 4 6 (6 ) , (6 ) Hence 4,( ) ( ) ( + 4 ) , 5 ( ) ( + 5 ) , ( ) ( ) Hence 8 5,( ) Theboundarycurvesintersectwhen +,. Thushere ( ) ( + ) , ( + 4 ) , ( ) ( ) Hence 9 4,( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

34 544 CHAPTER 5 MULTIPLE INTEGRALS 9. Notethatsin() for. sin() sin () sin(), 4 4 sin() sin () sin() cos() sin() integrate by parts with sin () sin() sin () Hence 4,( ) 8 4 [substitute cos()] cos () sin() cos() cos () (9) sin()]. ( ) 7 7, 4 ( 5 ) ( ) ( 7 ) 9 9 9, ( 4 ) ( 9 ) Hence 4,( ) ( ) sin, sin sin cos, sin cos sin cos cos, sin sin +sin Hence( ) ( ) ( + ),, 8 4 cos cos sin, c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

35 4 sin sin cos Hence( ) SECTION 5.5 APPLICATIONS OF DOUBLE INTEGRALS 545. ( ) +, ( ) () 7, ( ) (cos )() cos sin 4 () [thisistobeexpectedastheregionanddensity function are symmetric about the y-axis] Hence( ) ( ) (sin )() sin cos 4 (+) Now ( ) +,so ( ) () ()(), ( ) (cos )() sin (), ( ) (sin )() Hence( ). cos (+). cos sin 5. Placingthevertexoppositethehypotenuseat(), ( ) ( + ). Then + + ( ) 4 4 ( ) By symmetry, ( + ) ( ) + ( ) ( )5 5 5 Hence( ) 5 5. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

36 546 CHAPTER 5 MULTIPLE INTEGRALS 6. ( ) sin cos 56 6 Bysymmetryof and (),,and [(sin ) ] 56 sin sin (4sin sin ) cos + 4 cos 56 6 Hence( ) ( ). 7. ( ) ( ) 4 4 ( ) , ( ) ( ) , and ( ) 8. ( sin )( ) sin ( sin), ( cos )( ) 6 cos (+sin), and AsinExercise5,weplacethevertexoppositethehypotenuseat()andtheequalsidesalongthepositiveaxes. ( + ) ( + 4 ) ( ) + 5 ( ) ( ) , ( + ) ( 4 + ) ( )+ ( ) and ,. Ifwefindthemomentsofinertiaaboutthe -and -axes,wecandetermineinwhichdirectionrotationwillbemoredifficult. (SeetheexplanationfollowingExample4.) Themomentofinertiaaboutthe -axisisgivenby ( ) (+) (+) 8 (+) () 587 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

37 SECTION 5.5 APPLICATIONS OF DOUBLE INTEGRALS 547 Similarly,themomentofinertiaaboutthe -axisisgivenby ( ) (+) (+) ( + ) Since,moreforceisrequiredtorotatethefanbladeaboutthe -axis.. ( ), ( ) [], and (areaofrectangle)sincethelaminaishomogeneous. Hence and.. Hereweassume, butnotethatwearriveatthesameresultsif or. Wehave ( ),so 4 4 ( 4 ), 4 ( ) 4 4, and. Hence 6 and Inpolarcoordinates,theregionis ( ),so (sin ) sin sin , (cos ) cos + sin , and () 4 sincethelaminaishomogeneous. Hence sin sin cos, sin sin ( cos )sin cos + cos 4, 9 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

38 548 CHAPTER 5 MULTIPLE INTEGRALS sin sin cos +sin +cos [by integrating by parts twice] ( 4). Then,so 9 and 4 4,so. 5. Therightloopofthecurveisgivenby {( ) cos, 4 4}. UsingaCAS,we find ( ) ( + ) 4 cos Then ( ) ( ) ( ) 95. The moments of inertia are cos cos (cos ) (sin ) cos cos 4 cos sin,so ( ) 4 cos (sin ) 4 cos 5 sin , ( ) 4 cos (cos ) 4 cos 5 cos ,and UsingaCAS,wefind ( ) 8 ( ). Then ( ) ( ) ( ) (5 6 ) ( ) 79 8( ) 79( ) 768( ). (56 ) and 79( ),so 768( ) Themomentsofinertiaare ( ) 4 6 ( ), ( ) 4 8 ( ), and + 6 ( ). 7. (a) ( )isajointdensityfunction,soweknow R ( ). Since ( ) outsidethe rectangle[] [],wecansay Then. R ( ) ( ) (+) + 4 and c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

39 (b) ( ) ( ) + SECTION 5.5 APPLICATIONS OF DOUBLE INTEGRALS 549 (+) or (c) ( + ) (( ) )where isthetriangularregionshownin the figure. Thus ( + ) ( ) (+) (a) ( ),so isajointdensityfunctionif ( ). Here, ( ) outsidethesquare[] [], R so ( ) 4 R. Thus, ( )isajointdensityfunction. (b) (i) Norestrictionisplacedon,so ( ) (ii) ( ) 4 6 (c) Theexpectedvalueof isgivenby R ( ) Theexpectedvalueof is R ( ) (4) (4) (a) ( ),so isajointdensityfunctionif ( ). Here, ( ) outsidethefirstquadrant,so R ( ) R (5+) 5 5 lim 5 lim lim 5 lim 5 lim ( 5 ) lim 5( ) () ()( ) (5)( ) Thus ( )isajointdensityfunction. (b) (i) Norestrictionisplacedon,so ( ) ( ) (5+) 5 lim lim 5 lim 5 lim 5 lim () ()( ) (5)( ) 887 ( 5 ) lim 5( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

40 55 CHAPTER 5 MULTIPLE INTEGRALS (ii) ( 4) (c) Theexpectedvalueof isgivenby 4 ( ) 4 (5+) () ()( ) (5)( 8 ) 5 4 ( )( 8 ) ( ) (5+) R 5 lim 5 lim Toevaluatethefirstintegral,weintegratebypartswith and 5 (orwecanuseformula96 inthetableofintegrals): (+) 5. Thus lim (+) 5 lim 5 lim () (+) 5 lim(5) () lim + 5 (5)() [byl Hospital srule] Theexpectedvalueof isgivenby ( ) (5+) R 5 lim 5 lim Toevaluatethesecondintegral,weintegratebypartswith and (oragainwecanuseformula96in thetableofintegrals)whichgives (+5). Then lim 5 lim 5(+5) lim ( 5 ) lim 5 (+5) 5 +5 ()() (5) lim 5 5 [by l Hospital s Rule]. (a) The lifetime of each bulb has exponential density function () if if If and arethelifetimesoftheindividualbulbs,then and areindependent,sothejointdensityfunctionisthe product of the individual density functions: ( ) 6 (+) if, otherwise c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

41 SECTION 5.5 APPLICATIONS OF DOUBLE INTEGRALS 55 Theprobabilitythatbothofthebulbsfailwithinhoursis ( ) ( ) 6 (+) (b) Nowweareaskedfortheprobabilitythatthecombinedlifetimesofboth bulbsishoursorless. Thuswewanttofind ( + ),or equivalently (( ) )where isthetriangularregionshowninthe figure. Then ( + ) ( ) 6 6 (+) (+) (a) Therandomvariables and arenormallydistributedwith 45,, 5,and. Theindividualdensityfunctionsfor and,then,are () () 5 5 (45) and (). Since and areindependent,thejointdensityfunctionistheproduct ( ) () () Then (4 5, 5) (45) () (45) 5() 5 ( ) (45) 5(). UsingaCASorcalculatortoevaluatetheintegral,weget (4 5, 5) 5. (b) (4( 45) +( ) ) (45) 5(),where istheregionenclosedbytheellipse 4( 45) +( ). Solvingfor gives ± 4( 45),theupperandlowerhalvesofthe ellipse,andthesetwohalvesmeetwhere [sincetheellipseiscenteredat(45)] 4( 45) 45±. Thus 45+ 5() (45) (45) (45) 5(). 4(45) UsingaCASorcalculatortoevaluatetheintegral,weget (4( 45) +( ) ) 6. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

42 55 CHAPTER 5 MULTIPLE INTEGRALS. Because and areindependent,thejointdensityfunctionforxavier sandyolanda sarrivaltimesistheproductofthe individual density functions: ( ) () () SinceXavierwon twaitforyolanda,theywon tmeetunless. Additionally,Yolandawillwaituptohalfanhourbutnolonger,sothey won tmeetunless. Thustheprobabilitythattheymeetis (( ) )where istheparallelogramshowninthefigure. The integralissimplertoevaluateifweconsider asatypeiiregion,so 5 if, otherwise (( ) ) ( ) ( (+) + ) ( ) Byintegrationby parts (orformula 96 in the Table of Integrals), this is ( 5 ) (+) ( 5 )( ). Thusthereisonlyabouta%chancetheywillmeet. Such is student life!. (a) If ( )istheprobabilitythatanindividualat willbeinfectedbyanindividualat,and isthenumberof infectedindividualsinanelementofarea,then ( ) isthenumberofinfectionsthatshouldresultfrom exposureoftheindividualat toinfectedpeopleintheelementofarea.integrationover givesthenumberof infectionsofthepersonat duetoalltheinfectedpeoplein. Inrectangularcoordinates(withtheoriginatthecity s center),theexposureofapersonat is ( ) (b) If (),then [ ( )] 6 9 ( ) +( ) For attheedgeofthecity,itisconvenienttouseapolarcoordinatesystemcenteredat. Thenthepolarequationfor thecircularboundaryofthecitybecomes cos insteadof,andthedistancefrom toapoint inthecity c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

43 SECTION 5.6 SURFACE AREA 55 isagain (seethefigure). So cos cos 6 + cos cos 4 cos sin cos + sin sin + 4 sin Thereforetheriskofinfectionismuchlowerattheedgeofthecitythaninthemiddle,soitisbettertoliveattheedge. 5.6 Surface Area. Here ( ) ++4and istherectangle[5] [4],sobyFormulatheareaofthesurfaceis () [ ( )] +[ ( )] () 6(5)() 5 6. ( ) 5and isthedisk + 9,sobyFormula () () +(5) + () ( ) 9. ( ) 6 whichintersectsthe -planeintheline+ 6,so isthetriangularregiongivenby ( ). Thus () () +() () ( ) ++ with,. ThusbyFormula, () +() +(4) (+6 ) 4 (6 ) , (9 ) 4 4 () +[(9 ) ] sin + 9 sin 4 sin 6. ( ) 4 and istheprojectionoftheparaboloid 4 ontothe -plane,thatis, ( ) + 4. So, () () +() + (4 +) 4( + ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

44 554 CHAPTER 5 MULTIPLE INTEGRALS 7. ( ) with + 4. Then () (+4 ) ( ) ( + )and {( ) }. Then, and () +( ) + ++ (++) (+) (+) 4 5 ( ) 4 5 (5 7 +) 9. ( ) with +,so, () (+)5 5 (+)5 + ( +). Giventhesphere + + 4,when,weget + so ( ) + and ( ) 4. Thus () [()(4 ) ] +[()(4 ) ] (4 ) +4 4 (+4) 4., ( ), ( ), () + + cos + cos cos cos cos sin sin ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

45 SECTION 5.6 SURFACE AREA 555. Tofindtheregion : + implies + 4 or. Thus or aretheplaneswherethe surfacesintersect. But implies + +( ) 4,so intersectstheupperhemisphere. Thus ( ) 4 or + 4. Therefore istheregioninsidethecircle + +( ) 4,that is, ( ) +. () [()(4 ) ] +[()(4 ) ] (+4) 4. ( ),,. Then () + 4 ( ) +( ) + Converting to polar coordinates we have (4 ) ( + ) ( + ) +. () usingacalculator. 4. ( ) cos( + ), sin( + ), sin( + ). () + 4 sin ( + )+4 sin ( + )+ Converting to polar coordinates gives + 4( + )sin ( + )+. () 4 sin ( )+ 4 sin ( )+ 4 sin ( )+ 47 usingacalculator. 5. (a) Themidpointsofthefoursquaresare 4 4, 4 4, 4 4,and 4 4. Here ( ) +,sothemidpointrule gives () [( )] +[ ( )] + () +() (b) ACASestimatestheintegraltobe () This agrees with the Midpoint estimate only in the first decimal place () +() c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

46 556 CHAPTER 5 MULTIPLE INTEGRALS 6. (a) With wehavefoursquareswithmidpoints,,,and. Since + +,the Midpoint Rule gives () (+) +(+) (b) Using a CAS, we have () +( +) +(+) aboutofthemidpointruleestimate ,so () + UsingaCAS,wehave 4 or ln ( ) +++ +,. Weusea CAS to calculate the integral () + + (+) andfindthat () +sinh or () +ln ( +) +(+) Thisiswithin +4+(+8) ln ln ( ) +,. WeuseaCAS(withprecisionreducedtofivesignificantdigits,tospeed up the calculation) to estimate the integral () ,andfindthat ().. Let ( ) + +. Then +, + (+ ) + (+ ). WeuseaCAS toestimate ( ) In ordertographonlythepartofthesurfaceabovethesquare,weuse ( ) asthe -rangeinourplotcommand. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

47 SECTION 5.7 TRIPLE INTEGRALS 557. Here ( ) + +, ( ), ( ), so () ().. Let betheupperhemisphere. Then ( ),so () [( ) ] +[( ) ] lim + lim lim lim ()(). Thusthesurfaceareaoftheentiresphereis4.. Ifweprojectthesurfaceontothe -plane,thenthesurfacelies above thedisk + 5inthe -plane. Wehave ( ) + and,adaptingformula,theareaofthesurfaceis () + 5 [ ( )] +[ ( )] + Convertingtopolarcoordinates cos, sin wehave () (4 +) (4 +) 6 4. Firstwefindtheareaofthefaceofthesurfacethatintersectsthepositive -axis. AsinExercise,wecanprojecttheface ontothe -plane,sothesurfacelies above thedisk +. Then ( ) andtheareais () [( )] +[ ( )] [bythesymmetryofthesurface] This integral is improper (when ), so () lim 4 lim 4 lim 4 Since the complete surface consists of four congruent faces, the total surface area is 4(4) 6. lim Triple Integrals c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

48 558 CHAPTER 5 MULTIPLE INTEGRALS. There are six different possible orders of integration. (+ ) (+ ) (+ ) (+ ) (+ ) (+ ) (+ ) + (+9) (+ ) + (+9) +9 (6+8) +8 (+ ) (+ ) (+ ) (6 +8) +8 (+ ) ( ) ( ) ( ) ln ln (+) + ln ( ) + (ln ln) ln + + ln( +) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

49 SECTION 5.7 TRIPLE INTEGRALS cos(+ + ) sin(++ ) [sin(+) sin(+)] cos(+)+cos(+) cos +cos+ cos cos 6 sin+ sin sin 6 sin cos ( cos ) 9.. sin ( sin ) cos ( ) + ( ) [integrate by parts] 4 4 tan () tan () 4 4 (4 ) Here {( ) },so sin sin cos tan [cos( )+] sin( ) cos( )+ [integrate by parts] +. Here {( ) ++},so sin sin (++) + + ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

50 56 CHAPTER 5 MULTIPLE INTEGRALS 4. isthesolidabovetheregionshowninthe -planeandbelowtheplane +. Thus, + (+) ( + ) + ( ) Here {( ) },so ( ) ( ) ( ) ( ) ( ) 6. Here {( ) },so ( ) ( + ) Theprojectionof onthe -planeisthedisk +. Usingpolar coordinates cos and sin,weget (4 +4 ) 8 ( 4 ) 8 ( 5 ) 8() (9 ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

51 SECTION 5.7 TRIPLE INTEGRALS Theplane+ + 4intersectsthe -planewhen ,so {( ), 4, 4 }and (4 ) 4 4 4(4 ) (4 ) (4 ) ( 8+8) Theparaboloidsintersectwhen ,thustheintersectionisthecircle + 4, 4. Theprojectionof ontothe -planeisthedisk + 4,so ( ) Let ( ) + 4. Thenusingpolarcoordinates cos and sin,wehave 8 + (8 ) (8 ) 4 4 (6 8) 6 (8 ). Theplane + intersectsthe -planeintheline,so ( ),, and ( ) Here ( ) 4 + 4,so sin 4 (4 +) 4 4 using trigonometric substitution or FormulaintheTableofIntegrals sin () sin () [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

52 56 CHAPTER 5 MULTIPLE INTEGRALS Alternatively, use polar coordinates to evaluate the double integral: 4 (5 ) 4 (5 sin ) 5 sin 8 sin + 8 cos. (a) Thewedgecanbedescribedastheregion ( ) +,, ( ),, Sotheintegralexpressingthevolumeofthewedgeis. (b) ACASgives. 4 (OruseFormulasand87fromtheTableofIntegrals.) 4. (a) Divide into8cubesofsize 8. With ( ) + +,themidpointrulegives (b) UsingaCASwehave + + estimateinpart(a)byabout.5%. 8[()+()+()+()+() ()+()+()] Thisdiffersfromthe 5. Here ( ) cos()and 8,sotheMidpointRulegives ( ) cos +cos +cos +cos 9 +cos +cos 9 +cos 9 7 +cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

53 SECTION 5.7 TRIPLE INTEGRALS Here ( ) and,sothemidpointrulegives ( ) {( ),, }, the solid bounded by the three coordinate planes and the planes,. 8. ( ) 4 thesolidboundedbythethreecoordinateplanes,theplane, andthecylindricalsurface If,, aretheprojectionsof onthe -, -,and -planes,then ( ), 4 ( ) 4, 4 4 ( ) 4, 4 4 ( ), 4 4 ( ) +4 4 [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

54 564 CHAPTER 5 MULTIPLE INTEGRALS Therefore Then ( ), 4, 4 4 ( ) 4, 4 4, 4 4 ( ), 4 4, ( ) 4, 4 4, ( ), 4 4, 4 4 ( ), , 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ). If,, aretheprojectionsof onthe -, -,and -planes,then {( ), } ( ) + 9 {( ), } c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

55 SECTION 5.7 TRIPLE INTEGRALS 565 Therefore ( ),, 9 9 ( ), 9 9, ( ), 9 9, ( ),, 9 9 and ( ) 9 ( ) 9 9 ( ) 9 9 ( ) ( ) ( ) ( ). If,,and aretheprojectionsof onthe -, -,and -planes,then ( ) 4 ( ) 4 ( ) 4, ( ) 4,and ( ) ( ) 4 4 [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

56 566 CHAPTER 5 MULTIPLE INTEGRALS Therefore ( ), 4, ( ) 4,, ( ) 4,, ( ), 4, ( ),, 4 ( ), 4 4, 4 Then ( ) 4 ( ) 4 ( ) 4 ( ) 4 4 ( ) 4 4 ( ) 4 ( ). If,,and aretheprojectionsof onthe -, -,and -planes,then {( ), } {( ), }, ( ), {( ), },and ( ), {( ), } Therefore ( ),, (+ ) ( ),, (+ ) ( ),, + {( ),, + } ( ),, + {( ),, + } c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

57 SECTION 5.7 TRIPLE INTEGRALS 567 Then ( ) (+) ( ) (+) ( ) + ( ) ( ) + ( ) ( ) + +. The diagrams show the projections of onthe -, -,and -planes. Therefore ( ) ( ) () ( ) ( ) ( ) ( ) 4. Theprojectionsof ontothe -and -planesareasinthe firsttwodiagramsandso ( ) ( ) ( ) ( ) Nowthesurface intersectstheplane inacurvewhoseprojectioninthe -planeis ( ) or. Sowemustsplituptheprojectionof onthe -planeintotworegionsasinthethirddiagram. For( ) in, andfor( )in,,andsothegivenintegralisalsoequalto ( ) + ( ) ( ) + ( ) 5. ( ) ( ) where {( ),, }. [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

58 568 CHAPTER 5 MULTIPLE INTEGRALS If,,and aretheprojectionsof onthe -, -and -planesthen {( ), } {( ), }, {( ), } {( ), },and {( ), } {( ), }. Thuswealsohave {( ),, } {( ),, } {( ),, } {( ),, } {( ),, }. Then ( ) ( ) ( ) ( ) ( ) ( ) 6. ( ) ( ) where {( ),, }. Noticethat isboundedbelowbytwodifferentsurfaces,sowemustsplittheprojectionof ontothe -planeintotwo regionsasintheseconddiagram. If,,and aretheprojectionsof onthe -, -and -planesthen Thuswealsohave {( ), } {( ), } {( ), } {( ), }, {( ), } {( ), },and {( ), } {( ), }. {( ),, } {( ),, } {( ),, } {( ),, } {( ),, } {( ),, } {( ),, }. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

59 SECTION 5.7 TRIPLE INTEGRALS 569 Then ( ) ( ) + ( ) ( ) + ( ) ( ) ( ) ( ) 7. Theregion isthesolidboundedbyacircularcylinderofradiuswithaxisthe -axisfor. Wecanwrite (4+5 ) 4 + 5,but ( ) 5 isanoddfunctionwith respectto. Since issymmetricalaboutthe -plane,wehave 5. Thus (4+5 ) 4 4 () 4 () (4) Wecanwrite ( +sin +) + sin +. But isanoddfunctionwithrespect to andtheregion issymmetricaboutthe -plane,so. Similarly, sin isanodd functionwithrespectto and issymmetricaboutthe -plane,so sin. Thus ( +sin +) 4 () () ( ) ++ ( ) (++) ( ) ( ) + + ( ) (++) (++) (++) ( ) Thusthemassis 79 andthecenterofmassis( ) ( ) 4 ( 4 ) 6, 5 4 ( ) ( ) [continued] c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

60 57 CHAPTER 5 MULTIPLE INTEGRALS 4 4( ) 4( ) ( ) ( 5 ) [theintegrandisodd] 4 (4 4 ) ( ) ( ) Thus,( ) ( + + ) ( + ) ( + ) bysymmetryof and ( ) Hence( ) ( ) ( ) 6 ( ) ( ) 4 ( ) ( ) 6 4 ( )4 4 ( )4 ( ) ( ) 5 ( )5 6 ( ) ( ) + ( )4 4 Hence( ) ( )5 ( + ) Bysymmetry, 5. ( ) ( + ) ( + ) + + ( + ) Bysymmetry, ( + )and ( + ) ( )4 ( )4 + 4 ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

61 45. ( + ) ( ) + ( + ) ( + ) + ( ) () ( + )( ) + ( + ) + ( + ) + ( ) + 4 () (a) (b) ( )where + SECTION 5.7 TRIPLE INTEGRALS 57 +, +,and +. (c) ( + ) + ( + ) 48. (a) (b) ( )where (c) 49. (a) , + +, + + ( + )(++ + ) (b) ( ) (c) (+++ ) + 4 (+++ ) (+++ ) (+++ ) ( + )(+++ ) (a) 9 ( + ) 56 5 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

62 57 CHAPTER 5 MULTIPLE INTEGRALS (b) ( ) where 9 ( + ) 75, 9 ( + ) , 9 ( + ) (c) 9 ( + ), (a) ( )isajointdensityfunction,soweknow R ( ). Herewehave R ( ) Thenwemusthave8 8. (b) ( ) 8 ( ) ( ) (c) ( + + ) (( ) )where isthesolidregioninthefirstoctantboundedbythecoordinateplanes andtheplane ++. Theplane ++ meetsthe -planeintheline +,sowehave ( + + ) ( ) ( ) [( + )+( ) + ] ( + ) +( ) ( ) (a) ( )isajointdensityfunction,soweknow R ( ). Herewehave R ( ) ( ) 5 lim 5 lim lim lim 5 lim Sowemusthave. 5 lim (5++) lim ( 5 ) lim 5( ) lim ( ) ()( ) (5)( ) ()( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

63 SECTION 5.7 TRIPLE INTEGRALS 57 (b) Wehavenorestrictionon,so ( ) ( ) (5++) (c) ( ) lim [by part(a)] ( 5 )(5 5 )() ( 5 )( ) 7 ( ) 5 5. () ave 54. () 5 5 ( 5 )( )( ) ( ) ( ) ( ) Then ave ( + ) 4 4 (5++). + ) ( ( ) ( + ) ( ) ( ) () (a) Thetripleintegralwillattainitsmaximumwhentheintegrand ispositiveintheregion andnegative everywhereelse. Forif containssomeregion wheretheintegrandisnegative,theintegralcouldbeincreasedby excluding from,andif failstocontainsomepart oftheregionwheretheintegrandispositive,theintegralcould beincreasedbyincluding in. Sowerequirethat + +. Thisdescribestheregionboundedbythe ellipsoid + +. (b) Themaximumvalueof ( ) occurswhen isthesolidregionboundedbytheellipsoid + +. Theprojectionof onthe -planeistheplanarregionboundedbytheellipse +,so ( ) ( ) ( ) ( ) ( ) and usingacas. ( ( ) ( ) ) ( ) ( ) ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

64 574 CHAPTER 5 MULTIPLE INTEGRALS DISCOVERY PROJECT Volumes of Hyperspheres Inthisprojectweuse todenotethe -dimensionalvolumeofan -dimensionalhypersphere.. Theinteriorofthecircleisthesetofpoints ( ),. So, substituting sin andthenusingformula64toevaluatetheintegral,weget sin (cos ) cos + sin 4. The region of integration is ( ). Substituting sin andusingformula64tointegratecos,weget sin cos 4 ( ) cos 4. Herewesubstitute sin and,later, sin. Because cos seemstooccurfrequentlyin thesecalculations,itisusefultofindageneralformulaforthatintegral. FromExercises49and5inSection7., wehave sin 5 ( ) 4 6 and andfromthesymmetryofthesineandcosinefunctions,wecanconcludethat cos cos + sin sin + 5 ( ) 4 6 sin (+) (+) () () Thus 4 ( ) cos ( ) 4 ( ) cos 4 4 cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

65 4. Byusingthesubstitutions Problem,wecanwrite SECTION 5.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 575 +cos andthenapplyingformulasandfrom cos cos cos cos ( ) ( ( ) ( ) ) ) ( ( ) even odd Bycancelingwithineachsetofbrackets,wefindthat 4 6 () 4 6! 5 7 ()() 5 7 ( )! ()! even odd 5.8 Triple Integrals in Cylindrical Coordinates. (a) FromEquations, cos 4cos 4, sin 4sin 4,,sothepointis inrectangularcoordinates. (b) cos, sin, and,sothepointis( )inrectangularcoordinates.. (a) cos 4 sin 4,,and, sothepointis()inrectangularcoordinates. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

66 576 CHAPTER 5 MULTIPLE INTEGRALS (b) cos cos, sin sin,and, sothepointis(cossin)inrectangularcoordinates.. (a) FromEquationswehave () + so ;tan andthepoint()isinthesecond quadrantofthe -plane,so 4 +;. Thus,onesetofcylindricalcoordinatesis 4. (b) () +( ) 6so 4;tan andthepoint isinthesecondquadrantofthe -plane,so +;. Thus,onesetofcylindricalcoordinatesis (a) + 6so 4;tan andthepoint isinthefirstquadrantofthe -plane,so 6 +;. Thus,onesetofcylindricalcoordinatesis 4 6. (b) 4 +() 5so 5;tan 4 andthepoint(4 )isinthefourthquadrantofthe -plane, so tan ;. Thus,onesetofcylindricalcoordinates is 5tan 4 + (5564). 5. Since 4 but and mayvary,thesurfaceisaverticalhalf-planeincludingthe -axisandintersectingthe -planeinthe half-line,. 6. Since 5, + 5andthesurfaceisacircularcylinderwithradius5andaxisthe -axis ( + )or4,sothesurfaceisacircularparaboloidwithvertex(4),axisthe -axis,and opening downward. 8. Since + and +,wehave( + )+ or + +,anellipsoidcenteredatthe originwithintercepts ±, ±, ±. 9. (a) Substituting + and cos,theequation + + becomes cos + or +cos. (b) Substituting cos and sin, theequation becomes (cos ) (sin ) (cos sin )or cos.. (a) Substituting cos and sin,theequation+ + 6becomescos +sin + 6or 6 (cos +sin ). (b) Theequation + canbewrittenas ( + )+ whichbecomes + or + in cylindrical coordinates. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

67 SECTION 5.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 577. and describeasolidcircularcylinderwith radius,axisthe -axis,andheight,but restricts thesolidtothefirstandfourthquadrantsofthe -plane,sowehave a half-cylinder.. + isaconethatopensupward. Thus istheregionabovethis coneandbeneaththehorizontalplane. restrictsthesolidtothatpartof this region in the first octant.. Wecanpositionthecylindricalshellverticallysothatitsaxiscoincideswiththe -axisanditsbaseliesinthe -plane. Ifwe usecentimetersastheunitofmeasurement,thencylindricalcoordinatesconvenientlydescribetheshellas6 7,,. 4. Incylindricalcoordinates,theequationsare and 5. The curveofintersectionis 5 or 5. Sowegraphthesurfaces incylindricalcoordinates,with 5. InMaple,wecanusethe coordscylindrical option in a regular plotd command. In Mathematica, we can use RevolutionPlotD or ParametricPlotD. 5. The region of integration is given in cylindrical coordinates by ( ),,. This representsthesolidregionabovequadrantsiandivofthe -planeenclosed bythecircularcylinder,boundedabovebythecircularparaboloid ( + ),andboundedbelowbythe -plane( ). (4 ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

68 578 CHAPTER 5 MULTIPLE INTEGRALS 6. The region of integration is given in cylindrical coordinates by {( ),, }. Thisrepresentsthe solidregionenclosedbythecircularcylinder,boundedabovebythe cone,andboundedbelowbythe -plane Incylindricalcoordinates, isgivenby{( ) 4 5 4}. So () 64 5 (9) Theparaboloid + intersectstheplane 4inthecircle + 4or 4,soin cylindricalcoordinates, isgivenby ( ) 4. Thus 4 () Theparaboloid 4 4 intersectsthe -planeinthecircle + 4or 4,soin cylindricalcoordinates, isgivenby ( ) 4. Thus (++ ) 4 (cos + sin + ) (cos +sin ) + 4 (4 4 )(cos +sin )+ (4 ) (cos +sin ) (4 ) 64 5 (cos +sin ) (sin cos ) ( )+ 64 ( ) Incylindricalcoordinates isboundedbytheplanes, cos + sin +5andthecylinders and,so isgivenby{( ) cos + sin +5}. Thus cos +sin +5 (cos ) ( cos )(cos + sin +5) 4 4 (cos +cos sin )+ 5 cos ( cos )[ ] (cos +cos sin )+ 5 (7 8)cos 65 4 cos +sin +5 ( (cos +cos sin )+5 cos ) (+cos)+cos sin + 95 cos sin sin + 95 sin 65 4 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

69 SECTION 5.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 579. Incylindricalcoordinates, isboundedbythecylinder,theplane,andthecone. So {( ) }and cos cos 4 cos 5 5 cos 5 cos 5 (+cos) 5 + sin 5. Incylindricalcoordinates isthesolidregionwithinthecylinder boundedaboveandbelowbythesphere + 4, so ( ) 4 4. Thusthevolumeis (4 ) 4 (8 ) 4. Incylindricalcoordinates, isboundedbelowbythecone andabovebythesphere + or. The coneandthesphereintersectwhen,so ( ) andthevolumeis [] ( ) (+ ) 4 4. Incylindricalcoordinates, isboundedbelowbytheparaboloid andabovebythesphere + or. Theparaboloidandthesphereintersectwhen + 4 ( +)( ),so ( ) andthevolumeis [] ( ) 4 4 ( + 4 ) (a) Theparaboloidsintersectwhen ,sotheregionofintegration is ( ) + 9. Then, in cylindrical coordinates, ( ) 6,, and c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

70 58 CHAPTER 5 MULTIPLE INTEGRALS (b) Forconstantdensity, 6 frompart(a). Sincetheregionishomogeneousandsymmetric, and 6 () 6 ( ) ((6 ) 4 ) () (4) 4 Thus( ) 4 6 (5). 6. (a) cos (b) cos cos cos ( ) ( cos ) ( sin ) ( sin ) sin ( cos ) 4 cos + cos ( 4) ToplotthecylinderandthesphereonthesamescreeninMaple,wecanusethesequenceofcommands sphere:plotd(,theta..*pi,phi..pi,coordsspherical): cylinder:plotd(cos(theta),theta-pi/..pi/,z-..,coordscylindrical): with(plots): displayd({sphere,cylinder}); In Mathematica, we can use spheresphericalplotd[,{phi,,pi},{theta,,pi}] cylinderparametricplotd[{(cos[theta])ˆ,cos[theta]*sin[theta],z}, Show[sphere,cylinder] {theta,-pi/,pi/},{z,-,}] 7. Theparaboloid 4 +4 intersectstheplane when 4 +4 or + 4. So,incylindrical coordinates, ( ) 4. Thus 4 4 ( 4 ) 6 8 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

71 SECTION 5.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 58 Sincetheregionishomogeneousandsymmetric, and Hence( ) Sincedensityisproportionaltothedistancefromthe -axis,wecansay ( ) +. Then 8 ( ) sin () Theregionofintegrationistheregionabovethecone +,or,andbelowtheplane. Also,wehave with 4 4 whichdescribesacircleofradiusinthe -planecenteredat(). Thus, (cos ) (cos ) 4 4 cos [sin ] (cos ) (cos ) Theregionofintegrationistheregionabovetheplane andbelowtheparaboloid 9. Also,wehave with 9 whichdescribestheupperhalfofacircleofradiusinthe -planecenteredat(). Thus, (a) Themountaincomprisesasolidconicalregion. Theworkdoneinliftingasmallvolumeofmaterial withdensity ()toaheight ()abovesealevelis ()(). Summingoverthewholemountainweget ()(). (b) Here isasolidrightcircularconewithradius 6,ft,height,4ft, anddensity () lbft atallpoints in. Weusecylindricalcoordinates: 4 () () (6,) (,4) 9 ft-lb + 4 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

72 58 CHAPTER 5 MULTIPLE INTEGRALS DISCOVERY PROJECT The Intersection of Three Cylinders. Thethreecylindersintheillustrationinthetextcanbe visualizedasrepresentingthesurfaces +, +,and +. Thenwesketchthesolid of intersection with the coordinate axes and equations indicated. Tobemoreprecise,westartbyfindingthe boundingcurvesofthesolid(showninthefirstgraph below)enclosedbythetwocylinders + and + : ± ± arethesymmetric equations,andthesecanbeexpressedparametricallyas, ± ±,. Nowthecylinder + intersectsthesecurvesattheeightpoints ± ± ±. The resulting solid has twelve curved faces boundedby edges whicharearcsofcircles,asshowninthethirddiagram. Eachcylinderdefinesfourofthetwelvefaces.. To find the volume, we split the solid into sixteen congruent pieces,oneofwhichliesinthepartofthefirstoctantwith 4. (Naturally,weusecylindricalcoordinates!) This piece is described by ( ) 4,, andso,substituting cos,thevolumeoftheentire solid is cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

73 DISCOVERY PROJECT THE INTERSECTION OF THREE CYLINDERS 58. Tographtheedgesofthesolid,weuseparametrized curvessimilartothosefoundinproblemforthe intersection of two cylinders. We must restrict the parameter intervals so that each arc extends exactly to the desired vertex. One possible set of parametric equations(with all sign choices allowed) is, ±, ±, ; ±, ±,, ; ±,, ±,. 4. Letthethreecylindersbe +, +,and +. If,thenthefourfacesdefinedbythecylinder + inproblemcollapseintoasingleface,asinthefirst graph. If,theneachpairofverticallyopposedfaces,definedbyoneoftheothertwocylinders,collapseintoa singleface,asinthesecondgraph. If,thentheverticalcylinderenclosesthesolidofintersectionoftheothertwo cylinderscompletely,sothesolidofintersectioncoincideswiththesolidofintersectionofthetwocylinders + and +,asillustratedinproblem. Ifweweretovary or insteadof,wewouldgetsolidswiththesameshape,butdifferentlyoriented. 95,, 5. If,thesolidlookssimilartothefirstgraphinProblem4. AsinProblem,wesplitthesolidintosixteencongruent pieces,oneofwhichcanbedescribedasthesolidabovethepolarregion ( ), 4 inthe -plane andbelowthesurface cos. Thus,thetotalvolumeis 6 4 cos. If and,wehaveasolidsimilarto the second graph in Problem 4. Its intersection withthe -planeisgraphedattheright. Againwe split the solid into sixteen congruent pieces, one of whichisthesolidabovetheregionshowninthe secondfigureandbelowthesurface cos. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

74 584 CHAPTER 5 MULTIPLE INTEGRALS Wesplittheregionofintegrationwheretheoutsideboundarychangesfromtheverticalline tothecircle + or. isarighttriangle,socos. Thus,theboundarybetween and is cos in polarcoordinates,or inrectangularcoordinates. Usingrectangularcoordinatesfortheregion andpolar coordinatesfor,wefindthetotalvolumeofthesolidtobe cos () cos If,thecylinder + completelyenclosestheintersectionoftheothertwocylinders,sothesolidof intersectionofthethreecylinderscoincideswiththeintersectionof + and + asillustratedin Exercise Itsvolumeis Triple Integrals in Spherical Coordinates. (a) FromEquations, sin cos 6sin 6 cos 6, sin sin 6sin 6 sin 6,and cos 6cos 6,sothepointis 6 rectangular coordinates. in (b) sin cos 4, sin sin,and 4 cos 4,sothepointis in rectangular coordinates.. (a) sin cos, sin sin, cos sothepointis()inrectangularcoordinates. (b) 4sin cos 4 4sin sin 4 4 6, 4 6, 4cos 4 sothepointis 6 6 inrectangular coordinates. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

75 SECTION 5.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 585. (a) FromEquationsand, + + +() +,cos,and cos sin sin() [since ]. Thussphericalcoordinatesare. (b) ++,cos 4,and cos sin are 4. 4 sin(4) 4 [since ]. Thussphericalcoordinates 4. (a) ,cos. Thussphericalcoordinatesare. 6,andcos 6 sin sin(6) (b) ++ 4,cos 4 6 [since ]. Thussphericalcoordinatesare,andcos 6 sin sin(6) 5. Since,thesurfaceisthetophalfoftherightcircularconewithvertexattheoriginandaxisthepositive -axis. 6. Since, + + 9andthesurfaceisaspherewithcentertheoriginandradius. 7. sin sin sin sin ( ) + 4. Therefore,thesurfaceisasphereofradius centeredat. 8. sin sin +cos 9 (sin sin ) +(cos ) Thusthesurfaceisacircular cylinderofradiuswithaxisthe -axis. 9. (a) sin cos, sin sin,and cos,sotheequation + becomes (cos ) (sin cos ) +(sin sin ) or cos sin. If 6,thisbecomescos sin. ( corresponds to the origin which is included in the surface.) There are many equivalent equations in spherical coordinates, suchastan,cos,cos,oreven,. 4 4 (b) + 9 (sin cos ) +(cos ) 9 sin cos + cos 9or sin cos +cos 9.. (a) + + ( + + ) (sin cos ) or sin cos. (b) ++ sin cos +sin sin +cos or (sin cos +sin sin +cos ). c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

76 586 CHAPTER 5 MULTIPLE INTEGRALS. 4representsthesolidregionbetweenandincludingthespheresof radiiand4,centeredattheorigin. restrictsthesolidtothat portiononorabovethecone,and furtherrestrictsthe solidtothatportiononortotherightofthe -plane.. representsthesolidregionbetweenandincludingthespheresof radiiand,centeredattheorigin. restrictsthesolidtothat portiononorabovethe -plane,and furtherrestrictsthesolid tothatportiononorbehindthe -plane.. representsthesolidsphereofradiuscenteredattheorigin. 4 restrictsthesolidtothatportiononorbelowthecone representsthesolidsphereofradiuscenteredattheorigin. Notice that + (sin cos ) +(sin sin ) sin. Then csc sin sin +,so csc restricts the solid to that portion on or inside the circular cylinder becausethesolidliesabovethecone. Squaringbothsidesofthisinequalitygives cos cos. Theconeopensupwardsothattheinequalityis cos,orequivalently 4. Insphericalcoordinatesthesphere + + is cos cos. cos becausethesolidliesbelowthesphere. Thesolidcanthereforebedescribedastheregionin sphericalcoordinatessatisfying cos, 4. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

77 SECTION 5.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES (a) Thehollowballisasphericalshellwithouterradius5cmandinnerradius4.5cm. Ifwecentertheballattheoriginof the coordinate system and use centimeters as the unit of measurement, then spherical coordinates conveniently describe the hollowballas45 5,,. (b) Ifwepositiontheballasinpart(a),onepossibilityistotakethehalfoftheballthatisabovethe -planewhichis describedby45 5,,. 7. The region of integration is given in spherical coordinates by {( ) 6}. Thisrepresentsthesolid regioninthefirstoctantboundedabovebythesphere andbelowbythecone 6. 6 sin 6 sin cos 6 8. The region of integration is given in spherical coordinates by (9) 9 4 {( ) }. Thisrepresentsthesolid regionbetweenthespheres and andbelowthe -plane. 9. Thesolid ismostconvenientlydescribedifweusecylindrical coordinates: ( ). Then ( ) (cos sin ).. Thesolid is most convenientlydescribedifwe usespherical coordinates: ( ). Then sin sin cos () 7 ( ) (sin cos sin sin cos ) sin. 4. Insphericalcoordinates, isrepresentedby{( ) 5 }. Thus ( + + ) 5 ( ) sin sin 5 cos,5 7 4, ()() 78, c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

78 588 CHAPTER 5 MULTIPLE INTEGRALS. Insphericalcoordinates, isrepresentedby ( ) (9 ). Thus 9 ( sin cos + sin sin ) sin (9 sin ) sin 5 5 sin sin 8sin 4 5 sin 8sin 4 ( 5 cos )sin 8cos cos cos Insphericalcoordinates, isrepresentedby{( ) }and + sin cos + sin sin sin cos +sin sin. Thus ( + ) ( sin ) sin sin 4 ( cos ) sin 5 cos + 5 cos () (4 ) 5 + () Insphericalcoordinates, isrepresentedby{( ) }. Thus (sin sin ) sin sin sin 4 ( cos ) sin ( cos) cos + cos + sin 5 (4) Insphericalcoordinates, isrepresentedby ( ) + +. Thus (sin cos ) sin sin cos ( cos) cos integratebypartswith, sin [sin ] 4 ( ) (sin cos )(sin sin )(cos ) sin sin cos sin cos sin4 sin c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

79 SECTION 5.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES Thesolidregionisgivenby ( ) 6 anditsvolumeis 6 [cos ] 6 [] 8. Ifwecentertheballattheorigin,thentheballisgivenby sin sin 6 + () {( ) }andthedistancefromanypoint( )intheballtothe center()is + +. Thustheaveragedistanceis () 4 sin 4 sin cos ()() (a) Since 4cos implies 4cos,theequationisthatofasphereofradiuswithcenterat(). Thus 4cos sin 4cos 6 cos (b) Bythesymmetryoftheproblem. Then Hence( ) (). sin 64 cos sin 4cos cos sin cos sin 64cos cos6. Insphericalcoordinates,thesphere + + 4isequivalentto andthecone + isrepresented by. Thus,thesolidisgivenby 4 ( ) 4 and 4 sin sin 4 cos 4 () 8 8. (a) Bythesymmetryoftheregion, and. Assumingconstantdensity, 8 (fromexample4). Then 4 cos (cos ) sin 4 sin cos 4 sin cos cos 4 4 cos 5 sin Thusthecentroidis( ) 6 cos6 4 4 () cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

80 59 CHAPTER 5 MULTIPLE INTEGRALS (b) AsinExercise, + sin and ( + ) 4 cos ( sin ) sin 4 sin 4 sin cos () 6 6 cos6 + 8 cos cos 5 cos 5 cos sin (a) Placingthecenterofthebaseat(), ( ) + + isthedensityfunction. So sin sin cos 4 4 ()() (b) Bythesymmetryoftheproblem. Then 4 sin cos sin cos sin 5 () Hence( ) 5. (c) ( sin )( sin ) sin cos + cos 6 () (a) Thedensityfunctionis ( ),aconstant,andbythesymmetryoftheproblem. Then sin cos 4 sin cos 8 4. Butthemassis (volumeof thehemisphere),sothecentroidis 8. (b) Placethecenterofthebaseat();thedensityfunctionis ( ). Bysymmetry,themomentsofinertiaabout anytwosuchdiameterswillbeequal,sowejustneedtofind : 5 5 ( sin ) (sin sin +cos ) (sin sin +sin cos ) 5 5 sin cos + cos + cos 5 5 sin sin ( )+ ( ) Placethecenterofthebaseat(),thenthedensityis ( ), aconstant. Then (cos ) sin cos sin cos By the symmetry of the problem, and 4 cos sin 5 5 cos sin 5 5 cos 5 5. Hence( ) 8 5. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

81 SECTION 5.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES Inspherical coordinates + becomes cos sin or 4. Then 4 sin 4 sin 4 Hence( ) sin cos cos , 4 andbysymmetry Placethecenterofthesphereat(),letthediameterofintersectionbealongthe -axis,oneoftheplanesbethe -plane andtheotherbetheplanewhoseanglewiththe -planeis. Theninsphericalcoordinatesthevolumeisgivenby 6 6 sin 6 sin () Incylindricalcoordinatestheparaboloidisgivenby andtheplaneby sin andtheyintersectinthecircle sin. Then sin sin 5 [usingacas] (a) Theregionenclosedbythetorusis{( ),, sin },soitsvolumeis sin sin sin4 sin+ sin (b) InMaple,wecanplotthetorususingthecommand plotd(sin(phi),theta..*pi, phi..pi,coordsspherical);. In Mathematica, use SphericalPlotD[Sin[phi], {phi,,pi},{theta,,pi}]. 9. Theregion ofintegrationistheregionabovethecone + andbelowthesphere + + inthefirst octant. Because isinthefirstoctantwehave. Theconehasequation 4 (asinexample4),so 4, and. Sotheintegralbecomes 4 (sin cos )(sin sin ) sin 4 sin sin cos 4 cos cos cos sin sin Theregionofintegrationisthesolidsphere + +,so,,and. Also + + ( + + ) cos,sotheintegralbecomes cos sin sin cos 5 sin Theregionofintegrationisthesolidsphere + +( ) 4orequivalently sin +(cos ) 4cos cos,so,,and c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

82 59 CHAPTER 5 MULTIPLE INTEGRALS 4cos. Also( + + ) ( ),sotheintegralbecomes 4cos sin sin 6 4cos 6 6 (496) cos 6 sin () 496 sin 496cos 6 7 cos7 4. Thesolidregionbetweenthegroundandanaltitudeof5km(5m)isgivenby ( ) Thenthemassoftheatmosphereinthis region is (699 97) sin 67 6 sin [cos ] ()() (675 6 ) (67 6 ) kg (675 6 ) 4 (67 6 ) 4 4. Incylindricalcoordinates,theequationofthecylinderis,. Thehemisphereistheupperpartofthesphereradius,center(),equation +( ),. InMaple,wecanusethecoordscylindricaloption in a regular plotd command. In Mathematica, we can use ParametricPlotD. 44. We begin by finding the positions of Los Angeles and Montréal in spherical coordinates, using the method described in the exercise: Montréal 96mi Los Angeles 96mi NowwechangetheabovetoCartesiancoordinatesusing cos sin, sin sin and cos togettwo positionvectorsoflength96mi(sincebothcitiesmustlieonthesurfaceoftheearth). Inparticular: Montréal:h i LosAngeles:h i To find the angle between these twovectors we use the dot product: h i h i (96) cos cos 86 6rad. Thegreatcircledistancebetweenthecitiesis 96(6) 464mi. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

83 SECTION 5.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES If isthesolidenclosedbythesurface + sin6 sin5,itcanbedescribedinsphericalcoordinatesas 5 ( ) + 5 sin6sin5. Itsvolumeisgivenby () +(sin6sin5)5 sin 6 99 [usingacas]. 46. The given integral is equal to lim sin lim sin. Nowuse integrationbypartswith, toget lim () lim (Notethat as byl Hospital srule.) 47. (a) Fromthediagram, cot to, to sin (oruse cot ). Thus sin cot sin cot sin ( ) cot sin sin cot lim cos +sin cos ( cos ) (b) Thewedgeinquestionistheshadedarearotatedfrom to. Letting volumeoftheregionboundedbythesphereofradius andtheconewithangle ( to ) andletting bethevolumeofthewedge,wehave ( ) ( ) ( ) ( cos ) ( cos ) ( cos )+ ( cos ) ( ) ( cos ) ( cos ) ( ) (cos cos ) Or: Showthat sin cot sin cot. (c) BytheMeanValueTheoremwith () thereexistssome with suchthat ( ) ( ) ( )( )or. Similarlythereexists with suchthat cos cos sin. Substitutingintotheresultfrom(b)gives ( )( )(sin ) sin. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

84 594 CHAPTER 5 MULTIPLE INTEGRALS APPLIED PROJECT Roller Derby. + (+ ),so Theverticalcomponentofthespeedis sin,so sin + + sin.. Solvingtheseparabledifferentialequation,weget + sin But when,so andwehave + (+ ) sin sin. (sin )+. + (sin ). Solvingfor when gives + 4. Assumethatthelengthofeachcylinderis. Thenthedensityofthesolidcylinderis moment of inertia(using cylindrical coordinates) is andso. ( + ),andfromformulas5.7.6,its 4 4 Forthehollowcylinder,weconsideritsentiremasstolieadistance fromtheaxisofrotation,so + isa constant. Weexpressthedensityintermsofmassperunitareaas,andthenthemomentofinertiaiscalculatedasa doubleintegral: ( + ),so. 5. Thevolumeofsuchaballis 4 ( ) 4 ( ),andsoitsdensityis 4 ( ). UsingFormula5.9.,weget ( + ) 4 ( ) 4 ( ) 4 ( ) ( sin )( sin ) (+sin )cos 5 5 [from the Table of Integrals] 4 ( ) ( 5 ) 5 ( ) ( 5 ) 5( ) Therefore ( 5 ). Since representstheinnerradius, correspondstoasolidball,and correspondsto 5( ) a hollow ball. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

85 SECTION 5. CHANGE OF VARIABLES IN MULTIPLE INTEGRALS Forasolidball,,so ( 5 ) lim 5( ). Forahollowball,,so 5 ( 5 ) lim 5( ) 5 lim [by l Hospital s Rule] Note: Wecouldinsteadhavecalculated ( )( ) lim 5 5( )(++ ) 5. Thustheobjectsfinishinthefollowingorder: solidball 5,solidcylinder,hollowball,hollow cylinder( ). 5. Change of Variables in Multiple Integrals. 5, +. ( ) TheJacobianis ( ) 5 5() ()() 6..,. ( ) ( ). sin, cos. ( ) ( ) sin cos cos sin sin cos sin cos or cos 4. +,. ( ) ( ) ,,. ( ) ( ) , +, +. ( ) ( ) + ( )+(4 ) +8 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

86 596 CHAPTER 5 MULTIPLE INTEGRALS 7. Thetransformationmapstheboundaryof totheboundaryoftheimage,sowefirstlookatside inthe -plane. is describedby,,so + and. Eliminating,wehave, 6. is thelinesegment,,so 6+and. Then 6+( ) 5, 6. isthelinesegment,,so +6and,giving + +, 6. Finally, 4 isthesegment,,so and, 6. Theimageof set istheregion showninthe -plane,aparallelogramboundedbythesefoursegments. 8. isthelinesegment,,so and (+ ). Since,theimageistheline segment,. isthesegment,,so and (+ ) +. Thustheimageis theportionoftheparabola + for. isthesegment,,so and. Theimage isthesegment,. 4 isdescribedby,,so and (+ ). The imageisthelinesegment,. Thus,theimageof istheregion boundedbytheparabola +,the -axis,andthelines,. 9. isthelinesegment,,so and. Since,theimageistheportionofthe parabola,. isthesegment,,thus and,so. Theimageis thelinesegment,. isthesegment,,so and. The imageisthesegment,. Thus,theimageof istheregion inthefirstquadrantboundedbytheparabola,the -axis,andtheline. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

87 SECTION 5. CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 597. Substituting, into + gives +,sotheimageof + isthe ellipticalregion +.. isaparallelogramenclosedbytheparallellines, +andtheparallellines,. The firstpairofequationscanbewrittenas,. Ifwelet thentheselinesaremappedtothe verticallines, inthe -plane. Similarly,thesecondpairofequationscanbewrittenas +, +, andsetting + mapstheselinestothehorizontallines, inthe -plane. Boundarycurvesaremappedto boundarycurvesunderatransformation,soheretheequations, +defineatransformation that maps inthe -planetothesquare enclosedbythelines,,, inthe -plane. Tofindthe transformation thatmaps to wesolve, +for, : Subtractingthefirstequationfromthesecond gives ( )andaddingtwicethesecondequationtothefirstgives + (+). Thusonepossibletransformation (therearemany)isgivenby ( ), (+).. Theboundariesoftheparallelogram arethelines 4 or4, or4, or +, +5or +. Setting 4 and +definesatransformation thatmaps inthe -planetothesquare enclosedbythelines,,, inthe -plane. Solving 4, + for and gives 5 ( ), + (+). Thusonepossible 5 transformation isgivenby ( ), (+). 5 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

88 598 CHAPTER 5 MULTIPLE INTEGRALS. isaportionofanannularregion(seethefigure)thatiseasilydescribedinpolarcoordinatesas ( ). Ifweconvertedadoubleintegralover topolarcoordinatestheresultingregion ofintegrationisarectangle(inthe -plane),sowecancreateatransformation herebyletting playtheroleof and the roleof. Thus isdefinedby cos, sin and mapstherectangle ( ) inthe -planeto inthe -plane. 4. Theboundariesoftheregion arethecurves or, 4or 4, or, 4or 4. Setting and definesatransformation thatmaps inthe -planetothesquare enclosedby thelines, 4,, 4inthe -plane. Solving, for and gives [since,,, areallpositive],. Thusonepossibletransformation isgivenby,. 5. ( ) ( ) and (+) (+) 5. Tofindtheregion inthe -planethat correspondsto wefirstfindthecorrespondingboundaryunderthegiventransformation. Thelinethrough()and()is whichistheimageof + (+) ;thelinethrough()and()is + whichisthe imageof(+)+(+) + ;thelinethrough()and()is whichistheimageof + (+). Thus isthetriangle, inthe -planeand ( ) ( 5) ( ) 5 6 ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

89 6. SECTION 5. CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 599 ( ) ( ) , (+)+8 ( ) 5. isaparallelogramboundedbythe lines 4, 4,+,+ 8. Since and +, istheimageoftherectangle enclosedbythelines 4, 4,,and 8. Thus ( ) ( ) ( ) ( ) (4+8) ( 5) (96 4) , 4 andtheplanarellipse9 +4 6istheimageofthedisk +. Thus + (4 )(6) sin (4 cos ) 4 cos 4 4 4() 4 6 4, + + andtheplanarellipse + istheimageofthedisk +. Thus ( ) ( ) ( + ) + ( + 4 ) 8 4,, istheimageoftheparabola, istheimageoftheparabola,andthehyperbolas, aretheimagesofthelines and respectively. Thus. Here ( ), so ( ) ln ln ln 4ln ln. and isthe imageofthesquarewithvertices(),(),(),and(). So 4. (a) ( ) ( ) andsince,, thesolidenclosedbytheellipsoidistheimageofthe ball + +. So ()(volumeoftheball) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

90 6 CHAPTER 5 MULTIPLE INTEGRALS (b) Ifweapproximatethesurfaceoftheearthbytheellipsoid ,thenwecanestimate thevolumeoftheearthbyfindingthevolumeofthesolid enclosedbytheellipsoid. Frompart(a),thisis 4 (678)(678)(656) 8 km. (c) Themomentofintertiaaboutthe -axisis + ( ),where isthesolidenclosedby + +. Asinpart(a),weusethetransformation,,,so ( ) ( ) and ( + )() ( sin cos + sin sin ) sin ( sin cos ) sin + ( sin sin ) sin sin cos 4 + sin sin 4 cos cos + sin cos cos sin () () istheregionenclosedbythecurves,, 4,and 4,soifwelet and 4 then istheimageoftherectangleenclosedbythelines, ( )and, ( ). Now () ( ) and ( 5 5 ) 5 5, so ( ) ( ) Thustheareaof,andtheworkdoneby the engine, is 5 5 () 5 ln 5()(ln ln ) 5()ln.. Letting and,wehave ( )and ( ) ( ). Then 5 5 ( ) and istheimageoftherectangleenclosedbythelines, 4,,and 8. Thus Letting +and,wehave (+)and ( ) ( ). Then ( ) theimageoftherectangleenclosedbythelines,,,and. Thus ln ln8. (+) ( ) 6 4 (6 7) and is c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

91 CHAPTER 5 REVIEW 6 5. Letting, +,wehave (+), ( ) ( ). Then ( ) and isthe imageofthetrapezoidalregionwithvertices(),(),(),and(). Thus cos + cos sin sin() sin 6. Letting,,wehave9 +4 +,,and ( ). Then ( ) and istheimageofthe 6 quarter-disk givenby +,,. Thus sin(9 +4 ) 6 sin( + ) 6 sin( ) cos ( cos) 4 7. Let +and +. Then + (+)and ( ). ( ) ( ). Now + +,and + +. istheimageofthesquare regionwithvertices(),( ),( ),and(). So Let + and,then,, vertices(),()and(). Thus ( ) and istheimageunder ofthetriangularregionwith ( ) (+) () () () () asdesired. 5 Review. (a) AdoubleRiemannsumof is,where istheareaofeachsubrectangleand isa samplepointineachsubrectangle. If ( ),thissumrepresentsanapproximationtothevolumeofthesolidthatlies abovetherectangle andbelowthegraphof. (b) ( ) lim (c) If ( ), ( ) representsthevolumeofthesolidthatliesabovetherectangle andbelowthesurface ( ). If takesonbothpositiveandnegativevalues, ( ) isthedifferenceofthevolumeabove but belowthesurface ( )andthevolumebelow butabovethesurface ( ). (d) Weusuallyevaluate ( ) asaniteratedintegralaccordingtofubini stheorem(seetheorem5..4). c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

92 6 CHAPTER 5 MULTIPLE INTEGRALS (e) TheMidpointRuleforDoubleIntegralssaysthatweapproximatethedoubleintegral ( ) bythedouble Riemannsum wherethesamplepoints arethecentersofthesubrectangles. (f) ave ( ) where ()istheareaof. (). (a) See() and() and the accompanying discussion in Section 5.. (b) See() and the accompanying discussion in Section 5.. (c) See(5) and the preceding discussion in Section 5.. (d) See(6) () in Section 5... Wemaywanttochangefromrectangulartopolarcoordinatesinadoubleintegraliftheregion ofintegrationismoreeasily describedinpolarcoordinates. Toaccomplishthis,weuse ( ) (cos sin ) where is givenby,. 4. (a) ( ) (b) ( ), ( ) (c) Thecenterofmassis( )where and. (d) ( ), ( ), ( + )( ) 5. (a) ( ) (b) ( ) and R ( ). ( ) (c) Theexpectedvalueof is R ( ) ;theexpectedvalueof is R ( ). 6. () [ ( )] +[ ( )] + 7. (a) ( ) lim (b) Weusuallyevaluate ( ) asaniteratedintegralaccordingtofubini stheoremfortripleintegrals (see Theorem 5.7.4). (c) See the paragraph following Example (d) See(5) and(6) and the accompanying discussion in Section 5.7. (e) See() and the accompanying discussion in Section 5.7. (f) See()andtheprecedingdiscussioninSection5.7. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

93 CHAPTER 5 REVIEW 6 8. (a) ( ) (b) ( ), ( ), ( ). (c) Thecenterofmassis( )where,,and. (d) ( + )( ), ( + )( ), ( + )( ). 9. (a) See Formula and the accompanying discussion. (b) See Formula 5.9. and the accompanying discussion. (c) Wemaywanttochangefromrectangulartocylindricalorsphericalcoordinatesinatripleintegraliftheregion of integration is more easily described in cylindrical or spherical coordinates or if the triple integral is easier to evaluate using cylindrical or spherical coordinates.. (a) ( ) ( ) (b) See(9) and the accompanying discussion in Section 5.. (c) See() and the accompanying discussion in Section 5... This is true by Fubini s Theorem.. False. + describestheregionofintegrationasatypeiregion. Toreversetheorderofintegration,we mustconsidertheregionasatypeiiregion: +.. True by Equation sin Therefore the statement is true. sin (),since sin isanoddfunction. 5. True. ByEquation5..5wecanwrite () () () (). But () thisbecomes () () (). 6. Thisstatementistruebecauseinthegivenregion, + sin( ) (+)(),so 4 7. True: + sin( ) 4 () () 9. 4 thevolumeunderthesurface + + 4andabovethe -plane thevolumeofthesphere () 6 () so c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

94 64 CHAPTER 5 MULTIPLE INTEGRALS 8. True. Themomentofinertiaaboutthe -axisofasolid withconstantdensity is ( + )( ) ( ). 9. Thevolumeenclosedbythecone + andtheplane is,incylindricalcoordinates, 6,sotheassertionisfalse.. Asshowninthecontourmap,wedivide into9equallysizedsubsquares,eachwitharea. Thenweapproximate ( ) byariemannsumwith andthesamplepointstheupperrightcornersofeachsquare,so ( ) ( ) [()+()+()+()+() + ()+()+()+()] Using the contour lines to estimate the function values, we have ( ) [ ] 64. AsinExercise,wehave and. Usingthecontourmaptoestimatethevalueof atthecenterofeach. 4. subsquare, we have ( ) [(55)+(55)+(55)+(55)+(55) + (55)+(55)+(55)+(55)] [ ] 44 (+ ) + (+4 ) ( ) cos( ) cos( ) cos( ) sin( ) sin ( 4 ) integrate by parts inthefirstterm 7. sin (sin ) sin ( ) sin sin cos c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

95 CHAPTER 5 REVIEW ( ) Theregion ismoreeasilydescribedbypolarcoordinates: {( ) 4, }. Thus ( ) 4 (cos sin ).. Theregion isatypeiiregionthatcanbedescribedastheregionenclosedbythelines 4, 4+, andthe -axis. Sousingrectangularcoordinates,wecansay {( ) 4 4 4} and ( ) ( ).. Theregionwhoseareaisgivenby sin is ( ) sin,whichistheregioncontainedinthe loopinthefirstquadrantofthefour-leavedrose sin.. Thesolidis ( ) whichistheregioninthefirstoctantonorbetweenthetwo spheres and.. cos( ) cos( ) cos( ) cos( ) sin( ) sin 4. ( ) ( ) ((+) 4 ) ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

96 66 CHAPTER 5 MULTIPLE INTEGRALS ln(+ ) 4 ln tan ln(+ ) tan ln tan ln 4 ln (8 ) (8 ) ( ) (cos ) cos sin ( ) ( ) ( + ) (+) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

97 CHAPTER 5 REVIEW ( ) ( ) ( ) ( ) ( ) ( cos )( sin )( ) ( ) 4 sin ( 5 7 ) ( cos4) sin ( ) ( )( ) ( + ) sin cos + cos 64 5 (sin ) ( +4 ) ( cos )( sin ) cos sin 6 4 cos ( +84) (4 ) (+) ( ) (). sin ( sin ) 6 8 sin 6] + c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

98 68 CHAPTER 5 MULTIPLE INTEGRALS. Usingthewedgeabovetheplane andbelowtheplane andnotingthatwehavethesamevolumefor as for (souse ),wehave 9 ( 9 ) Theparaboloidandthehalf-coneintersectwhen + +,thatiswhen + or. So ( ) 4 () (a) ( ) 4 4 (b) ( ) ( ), ( 4 ). Hence( ) (c) ( 5 ), ( ) ( 4 ) 4, 4 + 8, 4,and (a) 4 where is constant, + cos cos,and sin [bysymmetry ]. Hencethecentroidis( ) 4 4. (b) 4 cos sin sin , 5 cos sin 8 sin , and 5 cos sin 4 sin Hence( ) (a) Theequationoftheconewiththesuggestedorientationis( ) +,. Then isthe volumeofonefrustumofacone;bysymmetry ;and + () + Hencethecentroidis( ) 4. (b) ()() ()() ( + ) ( 4 ) ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

99 CHAPTER 5 REVIEW Let ( ) + 4. ( ) +,so ( ) ( + ), ( ) ( + ),and + () () + 9. Let representthegiventriangle;then canbedescribedastheareaenclosedbythe -and -axesandtheline, orequivalently {( ), }. Wewanttofindthesurfaceareaofthepartofthegraphof + thatliesover,sousingequation5.6.wehave () + + +() +() ( ) UsingFormulaintheTableofIntegralswith,,and,wehave ln Ifwesubstitute +4 inthesecondintegral,then 8 and UsingFormula5.6.with sin, 4 6 (+4 ). Thus () +4 +ln (+4 ) 6+ln (6) ln + ln ln + + cos,weget sin + cos ( + ) 9 9 ( + ) 9 (cos )( ) cos 4 sin (4) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

100 6 CHAPTER 5 MULTIPLE INTEGRALS 4. Theregionofintegrationisthesolidhemisphere + + 4, sin (sin sin ) sin sin sin (+sin )cos Fromthegraph,itappearsthat at 7andat,with on(7). Sothedesiredintegralis 7 [( 7 ) ] Letthetetrahedronbecalled. Thefrontfaceof isgivenbytheplane + +,or, whichintersectsthe -planeintheline. Sothetotalmassis ( ) ( + + ) 7 5. Thecenterofmassis ( ) ( ) ( ) ( ) (a) ( )isajointdensityfunction,soweknowthat R ( ). Since ( ) outsidetherectangle [] [],wecansay R ( ) Then5 5. ( ) (+) + (+) + 5 (b) ( ) 5 ( ) ( ) (c) ( + ) (( ) )where isthetriangularregionshownin the figure. Thus ( + ) ( ) 5 (+) ( )+ ( ) ( ) 45 c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

101 CHAPTER 5 REVIEW Each lamp has exponential density function if () 8 8 if If,,and arethelifetimesoftheindividualbulbs,then,,and areindependent,sothejointdensityfunctionisthe product of the individual density functions: ( ) 8 (++)8 if,, otherwise Theprobabilitythatallthreebulbsfailwithinatotalofhoursis ( + + ),orequivalently (( ) )where isthesolidregioninthefirstoctantboundedbythecoordinateplanesandtheplane ++. Theplane ++ meetsthe -planeintheline +,sowehave ( + + ) ( ) (++)8 8 8 (++)8 8 [ 54 (+)8 ] (+)8 8 8 [ 54 (8 ) 8 8 ] 8 54 (8 ) (8) (8) ( ) ( ) 48. ( ) ( ) where ( ),,. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

102 6 CHAPTER 5 MULTIPLE INTEGRALS If,, and aretheprojections of onthe -, -, and -planes, then ( ), {( ) 8, }, {( ) 4, } ( ),, {( ) 8, 4}. Therefore we have ( ) ( ) Since and +, (+)and ( ). ( ) Thus ( ) and ( ) ( ) ( ) ( ) + ( ) ( ) 4 4 ln. 5. ( ) ( ) 8,so 8 4( ) 8( ) +4 ( ) 4 8 ( )4 + ( ) 4 4( ) ( )4 ( ) 4 ( ) 5 ( 5 )5 + ( ) Let and + so ( ) ( )and ( ) (+ ). ( ) ( ). istheimageunderthistransformationofthesquare withvertices( ) (),(),(),and(). So Thisresultcouldhavebeenanticipatedbysymmetry,sincetheintegrandisanoddfunctionof and issymmetricabout the -axis. 5. BytheExtremeValueTheorem(4.7.8), hasanabsoluteminimumvalue andanabsolutemaximumvalue in. Then byproperty5.., () ( ) (). Dividingthroughbythepositivenumber (),weget ( ). Thissaysthattheaveragevalueof over liesbetween and. But iscontinuous () on andtakesonthevalues and,andsobytheintermediatevaluetheoremmusttakeonallvaluesbetween and. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

103 Specifically,thereexistsapoint( )in suchthat ( ) () ( ) ( ) (). CHAPTER 5 REVIEW 6 ( ) orequivalently 5. Foreach suchthat lieswithinthedomain, ( ),andbythemeanvaluetheoremfordoubleintegralsthere exists( )in suchthat ( ) ( ). But lim ) ( ), +( so lim ( ) lim ( ) ( )bythecontinuityof (a) ( + ) ( ) ( ) if 6 ln() if (b) Theintegralinpart(a)hasalimitas + forallvaluesof suchthat. (c) ( + + ) ( ) sin sin 4 4 ( ) if 6 4ln() if (d) As +,theaboveintegralhasalimit,providedthat. c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

104

105 PROBLEMS PLUS. Let 5,where {( ) }. [+] 5 [+] 5 [+]],since [+] constant +for( ). Therefore [+] 5 (+)[( )] ( )+4( )+5( )+6( 4 )+7( 5 ) +4 +5() Let {( ), }. For,max if,. ave andmax if. Thereforewedivide intotworegions:,where {( ), }and {( ), }. Nowmax for ( ),andmax for( ) max{ } max{ } max{ } + () + cos( ) max{ } + cos( ) cos( ) [changingtheorderofintegration] cos( ) sin sin 4. Let a r, b r, c r,wherea h i,b h i,c h i. Underthischangeofvariables, correspondstotherectangularbox,,. So,byFormula5.., (a r)(b r)(c r) ( ) ( ). But ( ) ( ) a b c (a r)(b r)(c r) a b c a b c () 8 a b c c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. 65

106 66 CHAPTER 5 PROBLEMS PLUS 5. Since,exceptat(),theformulaforthesumofageometricseriesgives (),so 6. Let () + + () (+) and +. Weknowtheregionofintegrationinthe -plane,sotofinditsimageinthe -planeweget and intermsof and,andthenusethemethodsofsection5.. + similarly + +,so +,and. isgivenby,,sofromtheequationsderivedabove,theimageof is :,,,thatis,,.similarly,theimageof is :,,the imageof is :,,andtheimageof 4 is 4:,. ( ) TheJacobianofthetransformationis ( ). Fromthediagram, weseethatwemustevaluatetwointegrals: oneovertheregion ( ), and the other over ( ), +. So 4 + (+) ( ) + + arctan arctan arctan (+) ( ) + arctan + Nowlet sin,so cos andthelimitschangetoand 6 (inthefirstintegral)and 6 and (inthe c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

107 CHAPTER 5 PROBLEMS PLUS 67 second integral). Continuing: 6 4 sin sin arctan cos sin sin + 6 sin arctan cos sin 6 cos sin cos ( sin ) 4 arctan + arctan cos cos 6 cos cos 6 sin 4 arctan(tan ) + arctan cos But(following the hint) Continuing: sin cos 4 6 cos sin sin sin cos sin sin cos 6 arctan(tan ) tan 6 [half-angleformulas] arctan tan (a) Since exceptat(),theformulaforthesumofageometricseriesgives (),so () (+) () + + (b) Since,exceptat(),theformulaforthesumofageometricseriesgives + (),so + () () () (+) + () () + () + Toevaluatethissum,wefirstwriteoutafewterms: Noticethat BytheAlternatingSeriesEstimationTheoremfromSection.5,wehave 6 7. Thiserrorofwillnotaffecttheseconddecimalplace,sowehave 9. + c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

108 68 CHAPTER 5 PROBLEMS PLUS 8. arctan arctan arctan lim arctan + 9. (a) cos, sin,. Then + + cos + + +sin cos + sin + cos sin Similarly sin + cos and ln ln cos + sin and + + sin + cos sin cos cos sin. So cos + sin + cos cos sin + + sin + cos sin cos + + (b) sin cos, sin sin, cos. Then + + sin cos cos sin + sin cos + sin sin + cos,and + + sin sin + cos sin sin cos + sin cos cos + sin cos sin + sin cos + sin sin + cos Similarly cos cos + cos sin sin,and + sin c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

109 cos sin cos sin cos cos CHAPTER 5 PROBLEMS PLUS 69 sin cos sin + cos cos + cos sin And sin sin + sin cos,while Therefore + + sin sin cos sin sin cos sin cos sin + sin sin cot sin + sin cos sin cos sin sin (sin cos )+(cos cos )+sin + (sin sin )+(cos sin )+cos + cos +sin + sin cos +cos cos sin cos cos sin + sin sin +cos sin sin sin sin sin Butsin cos +cos cos sin cos cos (sin +cos ) cos andsimilarlythecoefficientof is.alsosin cos +cos cos +sin cos (sin +cos )+sin,andsimilarlythe coefficientof is. SoLaplace sequationinsphericalcoordinatesisasstated.. (a) Considerapolardivisionofthedisk,similartothatinFigure5.4.4,where,,andwherethepolarsubrectangle,aswellas,, and arethesameasinthat figure. Thus. Themassof is,anditsdistancefrom is ( ) +. Accordingto Newton slawofgravitation,theforceofattractionexperiencedby duetothispolarsubrectangleisinthedirection from towards andhasmagnitude. Thesymmetryofthelaminawithrespecttothe -and -axesandthe positionof aresuchthatallhorizontalcomponentsofthegravitationalforcecancel,sothatthetotalforceissimplyin the -direction. Thus,weneedonlybeconcernedwiththecomponentsofthisverticalforce;thatis, where istheanglebetweentheorigin, andthemass. Thussin andthepreviousresultbecomes sin, c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.

110 6 CHAPTER 5 PROBLEMS PLUS. ThetotalattractiveforceisjusttheRiemannsum ( ) ( ) + which becomes ( + as and. Therefore, ) ( + ) (b) Thisisjusttheresultofpart(a)inthelimitas. Inthiscase. + +,andweareleftwith +. () (),where {( ),, }. Ifwelet betheprojectionof onthe -planethen {( ), }. Andweseefromthediagram that {( ),, }. So () () () ( ) () + () + () + () ( ) (). + + Riemannsumofthefunction ( ) canbeconsideredadouble ++ wherethesquareregion {( ), }is dividedintosubrectanglesbydividingtheinterval[]onthe -axisinto subintervals,eachofwidth,and[]onthe -axisisdividedinto subintervals,eachofwidth. Thentheareaofeachsubrectangleis,andifwetakethe upperrightcornersofthesubrectanglesassamplepoints,wehave( ). Finally,notethat as,so lim + + lim + + lim ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.

111 ButbyDefinition5..5thisisequalto ( ),so lim + + ( ) (++) CHAPTER 5 PROBLEMS PLUS (+) (+) 4 ( +) 4 ( 4 +) Thevolumeis where isthesolidregiongiven. FromExercise5..(a),thetransformation,, mapstheunitball + + tothesolidellipsoid + + ( ) with ( ). Thesametransformationmapsthe plane ++ to + +. Thustheregion in -space correspondstotheregion in -spaceconsistingofthesmallerpieceofthe unitballcutoffbytheplane ++,a capofasphere (seethefigure). Wewillneedtocomputethevolumeof,butfirstconsiderthegeneralcase whereahorizontalplaneslicestheupperportionofasphereofradius toproduce acapofheight. Weusesphericalcoordinates. Fromthefigure,alinethroughthe originatangle fromthe -axisintersectstheplanewhencos ( ) ( )cos,andthelinepassesthroughtheouterrimofthecapwhen cos ( ) cos (( )). Thusthecap isdescribedby ( ) ( )cos cos (( )) anditsvolumeis cos (()) cos (()) cos (()) ()cos sin sin ()cos sin ( ) cos sin cos ( ) cos cos (()) ( ) + + ( ) ( ) ( )() ( ) c c Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, licated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpar to a publicly accessible in whole in part.