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}} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 286 38 " Math 132 \n Fall 2007 Exam III" }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 63 "1. The region in the first quadrant that is b ounded above by " }{XPPEDIT 602 0 "y = 4*x-x^2;" "6#/%\"yG,&*&\"\"%\" \"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 619 23 " and bound ed below by " }{XPPEDIT 620 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT 621 38 " is rotated about the vertical line " }{XPPEDIT 622 0 "x = -1;" "6# /%\"xG,$\"\"\"!\"\"" }{TEXT 623 58 ". What is the volume of the result ing solid of revolution?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 608 4 "a) " }{XPPEDIT 270 0 "15*Pi/2;" "6#*(\"#:\"\"\"%# PiGF%\"\"#!\"\"" }{TEXT 609 11 " b) " }{XPPEDIT 271 0 "16*Pi;" "6#*&\"#;\"\"\"%#PiGF%" }{TEXT 610 13 " c) " }{XPPEDIT 272 0 "18*Pi;" "6#*&\"#=\"\"\"%#PiGF%" }{TEXT 611 11 " d) " } {XPPEDIT 273 0 "21*Pi;" "6#*&\"#@\"\"\"%#PiGF%" }{TEXT 612 11 " \+ e) " }{TEXT 607 1 " " }{XPPEDIT 274 0 "45*Pi/2;" "6#*(\"#X\"\"\"%#PiG F%\"\"#!\"\"" }{TEXT 606 19 " " }}{PARA 0 "" 0 "" {TEXT 604 5 "f) " }{XPPEDIT 275 0 "20*Pi;" "6#*&\"#?\"\"\"%#PiGF%" } {TEXT 613 11 " g) " }{XPPEDIT 276 0 "27*Pi/2;" "6#*(\"#F\"\"\"% #PiGF%\"\"#!\"\"" }{TEXT 614 4 " " }{TEXT 617 3 " " }{TEXT 605 5 "h) " }{XPPEDIT 278 0 "25*Pi/2;" "6#*(\"#D\"\"\"%#PiGF%\"\"#!\"\"" } {TEXT 615 11 " i) " }{XPPEDIT 279 0 "14*Pi;" "6#*&\"#9\"\"\"%#P iGF%" }{TEXT 616 12 " j) " }{XPPEDIT 280 0 "15*Pi;" "6#*&\"#: \"\"\"%#PiGF%" }{TEXT 618 3 " " }}{PARA 3 "" 0 "" {TEXT 603 1 " " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 494 12 "Solution: e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "First we plot the planar region that is to be rotated." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> 4*x-x^2; \ng \+ := x -> x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(,&*&\"\"%\"\"\"9$F/F/*$)F0\"\"#F/!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(9$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " The planar region bounded by " }{XPPEDIT 18 0 "y = x;" "6#/% \"yG%\"xG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y = 4*x-x^2;" "6#/% \"yG,&*&\"\"%\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 21 " is shown b elow: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " " } {GLPLOT2D 629 382 382 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$\"\"!F)F(7$$\"_q +++++++++++++++++++++++++++++++++++++++++++++]i9Rl!$,\"$\"_q++++++++++ 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b]wFa^z7%F`]wFb]wFd^z7%F`]wFb]wFg^z7%F`]wFb]wFj^z7%F`]wFb]wF]_z7%F`]wF b]wF`_z7%F`]wFb]wFc_z7%F`]wFb]wFf_z7%F`]wFb]wFi_z-Febw6#%,PATCHNOGRIDG -%&TITLEG6$QYCut-away~of~The~Solid~of~Revolution~with~One~Shell~Shown6 \"-%%FONTG6%%&TIMESG%%BOLDG\"#9-%+PROJECTIONG6%$!#IF/$\"#vF/Fb[z" 1 2 0 1 10 0 2 1 1 1 2 1.000000 75.000000 -30.000000 1 0 "Curve 1" "Curve \+ 2" "Curve 3" }}}{PARA 0 "" 0 "" {TEXT -1 54 " \+ " }{TEXT 696 19 "Problem 1, Figure 3" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Here is the calculation of the \+ volume by means of the Method of Cylindrical Shells:" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "radius_of _shell := x -> x + 1;\nheight_of_shell := x -> f(x) - g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0radius_of_shellGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,&9$\"\"\"F.F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0h eight_of_shellGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"fG6#9$\"\"\"-% \"gGF/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Volume = Int(2*Pi*radius_of_shell(x)*height_of_shell(x), x = 0 .. 3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG-%$IntG6$,$**\"\"#\"\"\"%#Pi GF+,&%\"xGF+F+F+F+,&*&\"\"$F+F.F+F+*$)F.F*F+!\"\"F+F+/F.;\"\"!F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Volume = 2*Pi*Int(expand(rad ius_of_shell(x)*height_of_shell(x)), x = 0 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$,(*&F'F()%\"xG F'F(F(*$)F0\"\"$F(!\"\"*&F3F(F0F(F(/F0;\"\"!F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "antiderivative := int(2*x^2-x^3+3*x,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,(*(\"\"#\"\"\"\"\"$ !\"\"%\"xGF)F(*&\"\"%F*F+F-F**(F)F(F'F*F+F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Volume = simplify(subs(x=3,antiderivative)-subs( x=0,antiderivative));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG#\" #X\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 703 50 "Calculation of the Volume by the Method \+ of Washers" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 173 "It is possible (but more difficult) to obtain the volume using washers. A glance at Figure 2 above indicates that we ha ve to divide the integration into two pieces: one for " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{XPPEDIT 18 0 "`` <= 3;" "6#1%!G\"\"$" } {TEXT -1 16 " and one for " }{XPPEDIT 18 0 "3 <= y;" "6#1\"\"$%\"yG " }{XPPEDIT 18 0 "`` <= 4;" "6#1%!G\"\"%" }{TEXT -1 58 ". Figures 4 an d 5 below show the two types of washer. For " }{XPPEDIT 18 0 "y < 3;" "6#2%\"yG\"\"$" }{TEXT -1 54 ", the outer radius of the washer extends to the curve " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 7 ". 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he~Solid~of~Revolution~with~a~Washer~at~ Height~~y~>~3F)-%%FONTG6%%&TIMESG%%BOLDG\"#9-%+PROJECTIONG6%$\"#?\"\"! $\"#qFF\"\"\"" 1 2 0 1 10 0 2 1 1 1 2 1.000000 70.000000 20.000000 1 0 "Curve 1" "Curve 2" "Curve 3" }}}{PARA 0 "" 0 "" {TEXT -1 76 " \+ \+ " }{TEXT 700 19 "Problem 1, Figure 5" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Volume = \+ Int(Pi*( (1+y)^2-(1+2-sqrt(4-y))^2), y = 0 .. 3)+Int(Pi*( (1+2+sqrt(4- y))^2-(1+2-sqrt(4-y))^2), y = 3 .. 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG,&-%$IntG6$*&%#PiG\"\"\",&*$),&F+F+%\"yGF+\"\"#F+F+*$) ,&\"\"$F+*$,&\"\"%F+F0!\"\"#F+F1F9F1F+F9F+/F0;\"\"!F5F+-F'6$*&F*F+,&*$ ),&F5F+F6F+F1F+F+F2F9F+/F0;F5F8F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Volume = int(Pi*( (1+y)^2-(1+2-sqrt(4-y))^2), y = 0 \+ .. 3)+int(Pi*( (1+2+sqrt(4-y))^2-(1+2-sqrt(4-y))^2), y = 3 .. 4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG,$*(\"#X\"\"\"\"\"#!\"\"%#Pi GF(F(" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "2. " }{TEXT 288 1 " \+ " }{TEXT 289 54 " The region in the first quadrant bounded above by \+ " }{XPPEDIT 19 1 "y = x;" "6#/%\"yG%\"xG" }{TEXT 586 19 " and bel ow by " }{XPPEDIT 19 1 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT 587 8 " for " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 " `` <= 1;" "6#1%!G\"\"\"" }{TEXT 624 97 " is rotated about the line \+ y = 2. What is the volume of the solid of revolution that results?" } }{PARA 3 "" 0 "" {TEXT 287 3 "a) " }{XPPEDIT 390 0 "8*Pi/15;" "6#*(\" \")\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 385 14 " b) " }{XPPEDIT 391 0 "3*Pi/5;" "6#*(\"\"$\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 386 16 " \+ c) " }{XPPEDIT 392 0 "2*Pi/3;" "6#*(\"\"#\"\"\"%#PiGF%\"\"$! \"\"" }{TEXT 387 14 " d) " }{XPPEDIT 393 0 "4*Pi/5;" "6#*(\" \"%\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 388 15 " e) " }{XPPEDIT 394 0 "14*Pi/15;" "6#*(\"#9\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 389 14 " \+ \nf) " }{XPPEDIT 399 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT 395 17 " g) " }{XPPEDIT 400 0 "3*Pi/8;" "6#*(\"\"$ \"\"\"%#PiGF%\"\")!\"\"" }{TEXT 396 15 " h) " }{XPPEDIT 401 0 "5*Pi/8;" "6#*(\"\"&\"\"\"%#PiGF%\"\")!\"\"" }{TEXT 397 16 " \+ i) " }{XPPEDIT 402 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%! \"\"" }{TEXT 398 15 " j) " }{XPPEDIT 404 0 "7*Pi/8;" "6#*( \"\"(\"\"\"%#PiGF%\"\")!\"\"" }{TEXT 403 3 " " }{TEXT 405 6 " \n " }{TEXT 495 0 "" }}{PARA 0 "" 0 "" {TEXT 496 16 "Solution: a " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The region (shown with blue boundary) is rotated about " } {XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 639 354 354 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>F+7$$\"3[LL$e9ui2%F-F/7 $$\"3nmmm\"z_\"4iF-F27$$\"3[mmmT&phN)F-F57$$\"3CLLe*=)H\\5!#=F87$$\"3g mm\"z/3uC\"F:F<7$$\"3%)***\\7LRDX\"F:F?7$$\"3]mm\"zR'ok;F:FB7$$\"3w*** \\i5`h(=F:FE7$$\"3WLLL3En$4#F:FH7$$\"3qmm;/RE&G#F:FK7$$\"3\")*****\\K] 4]#F:FN7$$\"3$******\\PAvr#F:FQ7$$\"3)******\\nHi#HF:FT7$$\"3jmm\"z*ev :JF:FW7$$\"3?LLL347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\"3w***\\7o7Tv$F:Fjn7$ $\"3'GLLLQ*o]RF:F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++vV&RY2aF:Fbp7$$\"39mm;zXu9cF:Fep7$$\"3l******\\y) )GeF:Fhp7$$\"3'*)***\\i_QQgF:F[q7$$\"3@***\\7y%3TiF:F^q7$$\"35****\\P! 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