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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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%'NORMALG-%+PROJECTIONG6%$\"#?F-$\"#bF-\"\"\"" 1 2 0 1 10 0 2 1 1 4 2 1.000000 54.000000 24.000000 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}{PARA 0 "" 0 "" {TEXT 705 50 "Calculation of the Volum e by the Method of Washers" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The outer radius is the distance f rom " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 4 " t o " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 10 ", namely " }{XPPEDIT 18 0 "2-x^2;" "6#,&\"\"#\"\"\"*$%\"xGF$!\"\"" }{TEXT -1 42 " . The inner radius is the distance from " }{XPPEDIT 18 0 "y = x;" "6 #/%\"yG%\"xG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\" #" }{TEXT -1 10 ", namely " }{XPPEDIT 18 0 "2-x;" "6#,&\"\"#\"\"\"%\" xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "volume := Pi*int((2-x^2)^2-(2-x)^2, x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'volumeG,$*(\"\")\" \"\"\"#:!\"\"%#PiGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 706 43 "Calculation of the Volume by the Method of " }{TEXT -1 18 "Cylindrical Shells" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "For a shell that has an edge at level " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 18 ", the radius is " }{XPPEDIT 18 0 " 2-y;" "6#,&\"\"#\"\"\"%\"yG!\"\"" }{TEXT -1 34 " and the height of th e shell is " }{XPPEDIT 18 0 "sqrt(y)-y;" "6#,&-%%sqrtG6#%\"yG\"\"\"F' !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(2*Pi*(2-y)*(sqrt(y)-y), y = 0..1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"\"#:!\"\"%#PiGF&F&" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 48 "3. Calculate the arc length of the graph of " }{XPPEDIT 519 1 "y = x^3/3+1/(4*x);" "6#/%\"yG,&*&%\"xG\"\"$F(!\"\"\"\"\"*&F*F**&\"\"% F*F'F*F)F*" }{TEXT 518 9 " for " }{XPPEDIT 520 1 "1/2 <= x;" "6#1* &\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 521 1 "`` <= 1;" "6#1%!G\"\"\"" } {TEXT 408 1 "." }{TEXT 407 3 " \n\n" }{TEXT 406 4 "a) " }{XPPEDIT 417 0 "5/12;" "6#*&\"\"&\"\"\"\"#7!\"\"" }{TEXT 409 11 " b) " } {XPPEDIT 418 0 "11/24;" "6#*&\"#6\"\"\"\"#C!\"\"" }{TEXT 410 13 " \+ c) " }{XPPEDIT 419 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 411 14 " d) " }{XPPEDIT 420 0 "13/24;" "6#*&\"#8\"\"\"\"#C!\"\" " }{TEXT 412 12 " e) " }{TEXT 309 1 " " }{XPPEDIT 421 0 "7/12; " "6#*&\"\"(\"\"\"\"#7!\"\"" }{TEXT 293 19 " " }} {PARA 0 "" 0 "" {TEXT 291 5 "f) " }{XPPEDIT 422 0 "5/8;" "6#*&\"\"& \"\"\"\"\")!\"\"" }{TEXT 413 14 " g) " }{XPPEDIT 423 0 "2/3; " "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 414 6 " " }{TEXT 424 3 " \+ " }{TEXT 292 5 "h) " }{XPPEDIT 425 0 "17/24;" "6#*&\"#<\"\"\"\"#C!\" \"" }{TEXT 415 12 " i) " }{XPPEDIT 426 0 "3/4;" "6#*&\"\"$\"\" \"\"\"%!\"\"" }{TEXT 416 16 " j) " }{XPPEDIT 427 0 "5/6;" "6#*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT 428 3 " " }}{PARA 3 "" 0 "" {TEXT 290 1 " " }}{PARA 0 "" 0 "" {TEXT 497 13 "Solution: d\n" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := x -> x^3/3 + 1/(4*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\" \"$F/*$)9$F0F/F/F/*&#F/\"\"%F/*&F/F/F3!\"\"F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(f(x),x)^2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$),&*$)%\"xG\"\"#\"\"\"F**&F*F**&\"\"%F*F'F*!\"\"F.F )F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "expand(1+diff(f(x),x )^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"#F%*$)%\"xG\"\"% F%F%*&F%F%*&\"#;F%F(F%!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn := 1+diff(f(x),x)^2 = (x^2+1/(4*x^2))^2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/,&\"\"\"F'*$),&*$)%\"xG\"\"#F'F'*&F'F'*&\"\"% F'F,F'!\"\"F2F.F'F'*$),&F+F'*&F'F'*&F1F'F,F'F2F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "testeq( eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "` arc length = `*Int((x^2+1/(4*x^2)), x=1/2..1) = int((x^2+1/(4*x^2)), x =1/2..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%.arc~length~=~G\"\"\" -%$IntG6$,&*$)%\"xG\"\"#F&F&*&F&F&*&\"\"%F&F,F&!\"\"F&/F-;#F&F.F&F&#\" #8\"#C" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 644 1 "4" }{TEXT -1 2 ". " } {TEXT 645 138 "The line segment with end points (2,3) and (6,0) is r otated about the y-axis. What is the surface area of the resulting con ical frustum?" }}{PARA 3 "" 0 "" {TEXT 646 3 "a) " }{XPPEDIT 262 0 "12 *Pi;" "6#*&\"#7\"\"\"%#PiGF%" }{TEXT 647 14 " b) " } {XPPEDIT 263 0 "16*Pi;" "6#*&\"#;\"\"\"%#PiGF%" }{TEXT 648 14 " \+ c) " }{XPPEDIT 264 0 "20*Pi;" "6#*&\"#?\"\"\"%#PiGF%" }{TEXT 649 17 " d) " }{XPPEDIT 265 0 "24*Pi;" "6#*&\"#C\"\"\"%#PiGF% " }{TEXT 650 17 " e) " }{XPPEDIT 659 0 "25*Pi;" "6#*&\"#D \"\"\"%#PiGF%" }{TEXT 658 15 " \nf) " }{XPPEDIT 271 0 "27*Pi ;" "6#*&\"#F\"\"\"%#PiGF%" }{TEXT 651 14 " g) " }{XPPEDIT 272 0 "28*Pi;" "6#*&\"#G\"\"\"%#PiGF%" }{TEXT 652 14 " h) " }{XPPEDIT 273 0 "32*Pi;" "6#*&\"#K\"\"\"%#PiGF%" }{TEXT 653 18 " \+ i) " }{XPPEDIT 274 0 "36*Pi;" "6#*&\"#O\"\"\"%#PiGF%" }{TEXT 654 17 " j) " }{XPPEDIT 276 0 "40*Pi;" "6#*&\"#S\"\"\"%#P iGF%" }{TEXT 655 3 " " }{TEXT 656 6 " \n" }{TEXT 657 0 "" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 660 13 "Solution: j\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " " } {OLE 1 194048 1 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: B:ry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::Z:Zy=:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::j::<:::::::::::::::::::vYxI:;Z::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The average circumfere nce of the frustum is " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#Pi GF%" }{TEXT -1 2 " " }{TEXT 707 1 "(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(2+6)/2;" "6#*&,&\"\"#\"\"\"\"\"'F&F&F%!\"\"" }{TEXT -1 1 " " }{TEXT 708 1 ")" }{TEXT -1 1 "," }{TEXT 711 1 " " }{TEXT -1 3 "or " } {XPPEDIT 18 0 "8*Pi;" "6#*&\"\")\"\"\"%#PiGF%" }{TEXT -1 0 "" }{TEXT 709 1 "." }{TEXT -1 22 " The slant length is " }{XPPEDIT 18 0 "sqrt(3 ^2+4^2);" "6#-%%sqrtG6#,&*$)\"\"$\"\"#\"\"\"F+*$)\"\"%F*F+F+" }{TEXT -1 7 ", or " }{XPPEDIT 18 0 "5;" "6#\"\"&" }{TEXT -1 86 ". The surfa ce area is the product of the average circumference and the slant leng th: " }{XPPEDIT 18 0 "40*Pi;" "6#*&\"#S\"\"\"%#PiGF%" }{TEXT 710 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 367 17 "5. The graph of " }{XPPEDIT 538 1 "y = 3*sqrt(x);" "6#/%\"yG* &\"\"$\"\"\"-%%sqrtG6#%\"xGF'" }{TEXT 535 3 " " }{TEXT 536 7 " for \+ " }{XPPEDIT 257 1 "4 <= x;" "6#1\"\"%%\"xG" }{XPPEDIT 258 1 "`` <= 10 ;" "6#1%!G\"#5" }{TEXT 537 87 " is rotated about the x-axis. What i s the surface area of the resulting figure? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 369 3 "a) " }{TEXT 523 2 " " } {XPPEDIT 19 1 "54*Pi;" "6#*&\"#a\"\"\"%#PiGF%" }{TEXT 524 12 " \+ b) " }{XPPEDIT 19 1 "60*Pi;" "6#*&\"#g\"\"\"%#PiGF%" }{TEXT 525 12 " \+ c) " }{XPPEDIT 19 1 "64*Pi;" "6#*&\"#k\"\"\"%#PiGF%" }{TEXT 526 13 " d) " }{XPPEDIT 19 1 "72*Pi;" "6#*&\"#s\"\"\"%#PiGF% " }{TEXT 527 10 " e)" }{TEXT 370 3 " " }{XPPEDIT 19 1 "80*Pi; " "6#*&\"#!)\"\"\"%#PiGF%" }{TEXT 528 1 " " }{TEXT 529 11 " \+ " }}{PARA 0 "" 0 "" {TEXT 368 5 "f) " }{XPPEDIT 19 1 "84*Pi;" "6#*& \"#%)\"\"\"%#PiGF%" }{TEXT 522 13 " g) " }{XPPEDIT 19 1 "94*P i;" "6#*&\"#%*\"\"\"%#PiGF%" }{TEXT 530 2 " " }{TEXT 531 9 " h) \+ " }{XPPEDIT 19 1 "102*Pi;" "6#*&\"$-\"\"\"\"%#PiGF%" }{TEXT 532 11 " \+ i) " }{XPPEDIT 19 1 "109*Pi;" "6#*&\"$4\"\"\"\"%#PiGF%" }{TEXT 533 12 " j) " }{XPPEDIT 19 1 "115*Pi;" "6#*&\"$:\"\"\"\"%#PiGF %" }{TEXT 534 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 498 13 "Solution: i\n" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := x -> 3*sqrt(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"$\"\" \"-%%sqrtG6#9$F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "`Surface area` = 2*Pi*Int(f(x)*sqrt(1+diff(f(x),x)^2), x = 4 .. 10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%-Surface~areaG,$*(\"\"#\"\"\"%#P iGF(-%$IntG6$,$**\"\"$F(F'!\"\"%\"xG#F(F',&\"\"%F(*&\"\"*F(F1F0F(F2F(/ F1;F4\"#5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Answer := \+ Pi*Int(3*(4*x+9)^(1/2),x = 4 .. 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'AnswerG*&%#PiG\"\"\"-%$IntG6$,$*&\"\"$F',&*&\"\"%F'%\"xGF'F'\"\"* F'#F'\"\"#F'/F1;F0\"#5F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Answer := student[changevar](u = 4*x+9, Answer, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG*&%#PiG\"\"\"-%$IntG6$,$*(\"\"$F'\"\"%! \"\"%\"uG#F'\"\"#F'/F0;\"#D\"#\\F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "`Surface area` = value(Answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%-Surface~areaG,$*&\"$4\"\"\"\"%#PiGF(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 661 99 "6. Let R be the trapezoidal region in \+ the first quadrant that is bounded above by the graph of " } {XPPEDIT 258 1 "y = 2*x;" "6#/%\"yG*&\"\"#\"\"\"%\"xGF'" }{TEXT 662 56 ", below by the x-axis, on the left by the vertical line " } {XPPEDIT 668 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 667 40 ", and on the r ight by the vertical line " }{XPPEDIT 664 0 "x = 2;" "6#/%\"xG\"\"#" } {TEXT 663 53 ". What is the moment of R about the vertical line " } {XPPEDIT 666 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT 665 55 " ? (As sume that R has a constant density equal to 1.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 670 6 "a) " }{XPPEDIT 276 0 " 20/3;" "6#*&\"#?\"\"\"\"\"$!\"\"" }{TEXT 680 17 " b) " } {XPPEDIT 265 0 "7;" "6#\"\"(" }{TEXT 671 22 " c) " } {XPPEDIT 266 0 "23/3;" "6#*&\"#B\"\"\"\"\"$!\"\"" }{TEXT 672 21 " \+ d) " }{XPPEDIT 277 0 "8;" "6#\"\")" }{TEXT 679 18 " \+ e) " }{XPPEDIT 267 0 "25/3;" "6#*&\"#D\"\"\"\"\"$!\"\"" } {TEXT 673 10 " \nf) " }{XPPEDIT 268 0 "9;" "6#\"\"*" }{TEXT 674 20 " g) " }{XPPEDIT 269 0 "32/3;" "6#*&\"#K\"\"\"\"\"$ !\"\"" }{TEXT 675 18 " h) " }{XPPEDIT 270 0 "11;" "6#\"# 6" }{TEXT 676 21 " i) " }{XPPEDIT 271 0 "35/3;" "6#*& \"#N\"\"\"\"\"$!\"\"" }{TEXT 677 19 " j) " }{XPPEDIT 272 0 "12;" "6#\"#7" }{TEXT 669 1 "\n" }{TEXT 678 0 "" }}{PARA 0 "" 0 "" {TEXT 681 13 "Solution: c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> 2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,$*&\"\"#\"\"\"9$F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Moment[x=-1] := Int((x+1)*f(x), x = 1 .. \+ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'MomentG6#/%\"xG!\"\"-%$IntG 6$,$*(\"\"#\"\"\",&F(F0F0F0F0F(F0F0/F(;F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Moment[x=-1] := int((x+1)*f(x), x = 1 .. 2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'MomentG6#/%\"xG!\"\"#\"#B\"\"$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 344 88 "7. Let R be the region in the first quadrant that is bounded ab ove by the graph of " }{XPPEDIT 540 1 "y = 4-x^2;" "6#/%\"yG,&\"\"% \"\"\"*$%\"xG\"\"#!\"\"" }{TEXT 539 79 " and below by the x-axis. What is the x-coordinate of the center of mass of R?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 347 1 "\n" }{TEXT 348 4 "a) " } {XPPEDIT 357 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 351 18 " \+ b) " }{XPPEDIT 358 0 "3/8;" "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 352 18 " c) " }{TEXT 547 1 " " }{XPPEDIT 256 0 "2/5;" "6 #*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 548 16 " d) " }{XPPEDIT 359 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 356 16 " e) \+ " }{XPPEDIT 360 0 "3/5;" "6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 350 2 " \+ " }{TEXT 349 20 " \n" }{TEXT 345 5 "f) " } {XPPEDIT 361 0 "5/8;" "6#*&\"\"&\"\"\"\"\")!\"\"" }{TEXT 355 18 " \+ g) " }{XPPEDIT 362 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT 354 12 " " }{TEXT 363 1 " " }{TEXT 346 5 "h) " } {XPPEDIT 257 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 546 17 " \+ i) " }{XPPEDIT 364 0 "5/4;" "6#*&\"\"&\"\"\"\"\"%!\"\"" } {TEXT 353 18 " j) " }{XPPEDIT 365 0 "3/2;" "6#*&\"\"$\" \"\"\"\"#!\"\"" }{TEXT 366 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 499 13 "Solution : h\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f : = x -> 4 - x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,&\"\"%\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "M := Int(delta*f(x), x = 0..2); #Th e mass in terms of the mass density" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"MG-%$IntG6$*&%&deltaG\"\"\",&\"\"%F**$)%\"xG\"\"#F*!\"\"F*/F/;\" \"!F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "M := value(M);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,$*(\"#;\"\"\"\"\"$!\"\"%&deltaG F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Moment[`x=0`] := In t(delta*x*f(x), x = 0..2); # Moment about y-axis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'MomentG6#%$x=0G-%$IntG6$*(%&deltaG\"\"\"%\"xGF-,&\" \"%F-*$)F.\"\"#F-!\"\"F-/F.;\"\"!F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Moment[`x=0`] := value(Moment[`x=0`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%'MomentG6#%$x=0G,$*&\"\"%\"\"\"%&deltaGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "x_bar := Moment[`x=0`]/M; # x-coordinate of center of mass" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&x_barG#\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{GLPLOT2D 590 360 360 {PLOTDATA 2 "6 '-%'CURVESG6%7S7$$\"\"!F)$\"\"%F)7$$\"39LLLL3VfV!#>$\"3;Q4GO&*4)*R!#<7 $$\"3'pmm;H[D:)F/$\"3?/_j&f`L*RF27$$\"3LLLLe0$=C\"!#=$\"3/&*Qko&yX)RF2 7$$\"3ILLL3RBr;F;$\"3#*ojBs(p?(RF27$$\"3Ymm;zjf)4#F;$\"3S&[tB$*ef&RF27 $$\"3=LL$e4;[\\#F;$\"3/4(zk#*ex$RF27$$\"3p****\\i'y]!HF;$\"3Fsokz^g:RF 27$$\"3,LL$ezs$HLF;$\"3E\\O'yw_\"*)QF27$$\"3_****\\7iI_PF;$\"3#*Qj(3)> ?fQF27$$\"3#pmmm@Xt=%F;$\"3-)ek.ShY#QF27$$\"3QLLL3y_qXF;$\"3'p^Kbv-6z$ F27$$\"3i******\\1!>+&F;$\"3#fHv))*)4)\\PF27$$\"3()******\\Z/NaF;$\"3? u\\l&)Gg/PF27$$\"3'*******\\$fC&eF;$\"3evfc&>([dOF27$$\"3ELL$ez6:B'F;$ \"37+RQ2Eo6OF27$$\"3Smmm;=C#o'F;$\"3g8f.VkZ`NF27$$\"3-mmmm#pS1(F;$\"3$ [NdRD*)4]$F27$$\"3]****\\i`A3vF;$\"3f<\"f!>bEOMF27$$\"3slmmm(y8!zF;$\" 3_\"pce8#ovLF27$$\"3V++]i.tK$)F;$\"3>1(eq/ccI$F27$$\"39++](3zMu)F;$\"3 e,XYMd^NKF27$$\"3#pmm;H_?<*F;$\"3eA$*enXteJF27$$\"3emm;zihl&*F;$\"3#oT (*>&)*)\\3$F27$$\"39LLL3#G,***F;$\"3RE1\"QQt>+$F27$$\"3\"HF27$$\"3!****\\PQ#\\\"3\"F2$\"3AV2*QAu.$GF27$$\"3BLL$e\"*[H 7\"F2$\"3AT(GCt&)*QFF27$$\"3#*******pvxl6F2$\"3*\\*[GdE'4k#F27$$\"3z** **\\_qn27F2$\"3%G\"\\'o8;:a#F27$$\"3%)***\\i&p@[7F2$\"3\"e)*H,Va>W#F27 $$\"3#)****\\2'HKH\"F2$\"3@'R&G#=dvK#F27$$\"3_mmmwanL8F2$\"3$*Rz$Hs48A #F27$$\"3'******\\2goP\"F2$\"3J%*4(QLcU5#F27$$\"3CLLeR<*fT\"F2$\"3wlF27$$\"3'******\\)Hxe9F2$\"33!>M#y8)>(=F27$$\"3Ymm\"H!o-*\\\"F2 $\"3#yVPTk=Hv\"F27$$\"3))***\\7k.6a\"F2$\"3z\"*f#pc**\\i\"F27$$\"3emmm T9C#e\"F2$\"3c)es--7l\\\"F27$$\"3\"****\\i!*3`i\"F2$\"33HWEf4Pe8F27$$ \"3QLLL$*zym;F2$\"3(*R)z_y<=A\"F27$$\"3GLL$3N1#4F2$\"3)>F+uTd`G$F;7$$\"3/++v.Uac>F2$\"3u%yAxyZ$>(>%F ;7$Fi[l$\"3Qmmm\">K'*)\\F;7$Fi[l$\"3P*****\\Kd,\"eF;7$Fi[l$\"3-mmm\"fX (emF;7$Fi[l$\"3.*****\\U7Y](F;7$Fi[l$\"3'QLLLV!pu$)F;7$Fi[l$\"3xmmm;c0 T\"*F;7$Fi[l$\"3#*******H,Q+5F27$Fi[l$\"3)*******\\*3q3\"F27$Fi[l$\"3) *******p=\\q6F27$Fi[l$\"3mmm;fBIY7F27$Fi[l$\"3GLLLj$[kL\"F27$Fi[l$\"3? LLL`Q\"GT\"F27$Fi[l$\"3!*****\\s]k,:F27$Fi[l$\"39LLL`dF!e\"F27$Fi[l$\" 33++]sgam;F27$Fi[l$\"3/++]F27$Fi[l$\"3immmTc-)*>F27$Fi[l$\"3Mmm;f`@'3#F27$Fi[l$\"3y****\\nZ )H;#F27$Fi[l$\"3YmmmJy*eC#F27$Fi[l$\"3')******R^bJBF27$Fi[l$\"3f***** \\5a`T#F27$Fi[l$\"3o****\\7RV'\\#F27$Fi[l$\"3k*****\\@fke#F27$Fi[l$\"3 /LLL`4NnEF27$Fi[l$\"3#*******\\,s`FF27$Fi[l$\"3[mm;zM)>$GF27$Fi[l$\"3$ *******pfa " 0 "" {MPLTEXT 1 0 71 "Moment[`y=0`] := Int(delta*f(x)^2/2, x = 0 .. 2); # Moment about x-axis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'MomentG6#%$y=0G-%$IntG6 $,$*(\"\"#!\"\"%&deltaG\"\"\",&\"\"%F0*$)%\"xGF-F0F.F-F0/F5;\"\"!F-" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Moment[`y=0`] := value(Mom ent[`y=0`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'MomentG6#%$y=0G,$* (\"$G\"\"\"\"\"#:!\"\"%&deltaGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "y_bar := Moment[`y=0`]/M; # y-coordinate of center of mass" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&y_barG#\"\")\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ " }{GLPLOT2D 375 375 375 {PLOTDATA 2 "6)-%'CURVESG 6%7S7$$\"\"!F)$\"\"%F)7$$\"39LLLL3VfV!#>$\"3;Q4GO&*4)*R!#<7$$\"3'pmm;H [D:)F/$\"3?/_j&f`L*RF27$$\"3LLLLe0$=C\"!#=$\"3/&*Qko&yX)RF27$$\"3ILLL3 RBr;F;$\"3#*ojBs(p?(RF27$$\"3Ymm;zjf)4#F;$\"3S&[tB$*ef&RF27$$\"3=LL$e4 ;[\\#F;$\"3/4(zk#*ex$RF27$$\"3p****\\i'y]!HF;$\"3Fsokz^g:RF27$$\"3,LL$ ezs$HLF;$\"3E\\O'yw_\"*)QF27$$\"3_****\\7iI_PF;$\"3#*Qj(3)>?fQF27$$\"3 #pmmm@Xt=%F;$\"3-)ek.ShY#QF27$$\"3QLLL3y_qXF;$\"3'p^Kbv-6z$F27$$\"3i** ****\\1!>+&F;$\"3#fHv))*)4)\\PF27$$\"3()******\\Z/NaF;$\"3?u\\l&)Gg/PF 27$$\"3'*******\\$fC&eF;$\"3evfc&>([dOF27$$\"3ELL$ez6:B'F;$\"37+RQ2Eo6 OF27$$\"3Smmm;=C#o'F;$\"3g8f.VkZ`NF27$$\"3-mmmm#pS1(F;$\"3$[NdRD*)4]$F 27$$\"3]****\\i`A3vF;$\"3f<\"f!>bEOMF27$$\"3slmmm(y8!zF;$\"3_\"pce8#ov LF27$$\"3V++]i.tK$)F;$\"3>1(eq/ccI$F27$$\"39++](3zMu)F;$\"3e,XYMd^NKF2 7$$\"3#pmm;H_?<*F;$\"3eA$*enXteJF27$$\"3emm;zihl&*F;$\"3#oT(*>&)*)\\3$ F27$$\"39LLL3#G,***F;$\"3RE1\"QQt>+$F27$$\"3\"H F27$$\"3!****\\PQ#\\\"3\"F2$\"3AV2*QAu.$GF27$$\"3BLL$e\"*[H7\"F2$\"3AT (GCt&)*QFF27$$\"3#*******pvxl6F2$\"3*\\*[GdE'4k#F27$$\"3z****\\_qn27F2 $\"3%G\"\\'o8;:a#F27$$\"3%)***\\i&p@[7F2$\"3\"e)*H,Va>W#F27$$\"3#)**** \\2'HKH\"F2$\"3@'R&G#=dvK#F27$$\"3_mmmwanL8F2$\"3$*Rz$Hs48A#F27$$\"3'* *****\\2goP\"F2$\"3J%*4(QLcU5#F27$$\"3CLLeR<*fT\"F2$\"3wlF27$ $\"3'******\\)Hxe9F2$\"33!>M#y8)>(=F27$$\"3Ymm\"H!o-*\\\"F2$\"3#yVPTk= Hv\"F27$$\"3))***\\7k.6a\"F2$\"3z\"*f#pc**\\i\"F27$$\"3emmmT9C#e\"F2$ \"3c)es--7l\\\"F27$$\"3\"****\\i!*3`i\"F2$\"33HWEf4Pe8F27$$\"3QLLL$*zy m;F2$\"3(*R)z_y<=A\"F27$$\"3GLL$3N1#4F2$\"3)>F+uTd`G$F;7$$\"3/++v.Uac>F2$\"3u%yAxyZ$>(>%F;7$Fi[l$\"3Q mmm\">K'*)\\F;7$Fi[l$\"3P*****\\Kd,\"eF;7$Fi[l$\"3-mmm\"fX(emF;7$Fi[l$ \"3.*****\\U7Y](F;7$Fi[l$\"3'QLLLV!pu$)F;7$Fi[l$\"3xmmm;c0T\"*F;7$Fi[l $\"3#*******H,Q+5F27$Fi[l$\"3)*******\\*3q3\"F27$Fi[l$\"3)*******p=\\q 6F27$Fi[l$\"3mmm;fBIY7F27$Fi[l$\"3GLLLj$[kL\"F27$Fi[l$\"3?LLL`Q\"GT\"F 27$Fi[l$\"3!*****\\s]k,:F27$Fi[l$\"39LLL`dF!e\"F27$Fi[l$\"33++]sgam;F2 7$Fi[l$\"3/++]F27$Fi[ l$\"3immmTc-)*>F27$Fi[l$\"3Mmm;f`@'3#F27$Fi[l$\"3y****\\nZ)H;#F27$Fi[l $\"3YmmmJy*eC#F27$Fi[l$\"3')******R^bJBF27$Fi[l$\"3f*****\\5a`T#F27$Fi [l$\"3o****\\7RV'\\#F27$Fi[l$\"3k*****\\@fke#F27$Fi[l$\"3/LLL`4NnEF27$ Fi[l$\"3#*******\\,s`FF27$Fi[l$\"3[mm;zM)>$GF27$Fi[l$\"3$*******pfa(z(F ;$\"3-twaMe4/;F27$$\"3?$pkH19-z(F;$\"3w%yyo')*f2;F27$$\"3%3$eO%fdux(F; $\"3=pX(4L49h\"F27$$\"3q0RcF@ffxF;$\"3pg$=GEP]h\"F27$$\"3@_i)4cFrt(F;$ \"3*)o+,dnP=;F27$$\"3**z\"f]AxCr(F;$\"3@xE5G'y6i\"F27$$\"3qz&*QsO\\$o( F;$\"3%Qc[r&RtB;F27$$\"3)HA*oYJK]wF;$\"3e2jTy?'fi\"F27$$\"31D,M*=/Yh(F ;$\"3?!=LlpCxi\"F27$$\"3eVmG)ehdd(F;$\"3>WMS/w-H;F27$$\"3\"RXyY5a.a(F; $\"3*4B\"4_tsH;F27$$\"3/zn!z'3#)*\\(F;$\"3c[>l%*****H;F27$$\"3SbFAHb7f uF;$\"3ZeG'=C?(H;F27$$\"3'\\G*[NXh?uF;$\"3*eaZwfI*G;F27$$\"3go!o?t1oQ( F;$\"3Toug*f#yF;F27$$\"37D8zT0t[tF;$\"32O#>_02fi\"F27$$\"3J=$4dJ:)=tF; $\"3^%>:,p5Ri\"F27$$\"3ysqf40K(G(F;$\"3t-6=;$e6i\"F27$$\"33BM$Q(Q(GE(F ;$\"3ENH:UpP=;F27$$\"30#p0yz?-C(F;$\"30![G7#\\+:;F27$$\"381f!=\">2BsF; $\"3*f%)f(es`6;F27$$\"3,$)yr%H\"45sF;$\"39W)z9`:xg\"F27$$\"3N!\\<%*4*y -sF;$\"3Hc^,v73/;F27$$\"3+-:sU9++sF;$\"3w%4$GRI4+;F27$$\"3<3$Q%fou-sF; $\"3w-9V/'\\ff\"F27$$\"3v$R#)f.y(4sF;$\"3[t^\"4:.Cf\"F27$$\"3A0!4O8-@A (F;$\"3gW5RH$)p)e\"F27$$\"3S^q.;WxRsF;$\"3[R1MFE2&e\"F27$$\"35-;'o'zhh sF;$\"3#3\"H>niy\"e\"F27$$\"3Egs)=-$o'G(F;$\"3]&zjl&f!*y:F27$$\"3Ukk42 Ca=tF;$\"39xlB9+6w:F27$$\"35tBXZ$z-N(F;$\"3wf^wHJ+u:F27$$\"3C#Q[bb;oQ( F;$\"35%fz,+^!G$o (F;$\"3?`W7y&\\id\"F27$$\"3xFSf6!zHr(F;$\"3QM#=Q$=()y:F27$$\"3]b;\"=N0 qt(F;$\"3C7[M%[2;e\"F27$$\"3w.j[D,BgxF;$\"3igA3>M2&e\"F27$$\"3CvkfG+vw xF;$\"3_y)pU4?%)e\"F27$$\"3#fowgmS'*y(F;$\"3%[(4[[W=#f\"F27$$\"3Wn*ygm 3sz(F;$\"37f:3)4 " 0 "" {MPLTEXT 1 0 22 "a := 2:\nf := x -> x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eqn := f(7) = simplify( int(f(x),x = a .. b)/(b-a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqn G/\"#\\,(*&\"\"$!\"\"%\"bG\"\"#\"\"\"*(F,F-F)F*F+F-F-#\"\"%F)F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn, b);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$!#8\"#6" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 641 2 "10" }{TEXT -1 1 "." }{TEXT 557 18 " Suppose that \+ " }{XPPEDIT 559 1 "f(x) = c/(1+x^2);" "6#/-%\"fG6#%\"xG*&%\"cG\"\" \",&F*F**$F'\"\"#F*!\"\"" }{TEXT 558 94 " is the probability densit y function of a random variable that has values in the interval " } {XPPEDIT 636 0 "[0, sqrt(3)];" "6#7$\"\"!-%%sqrtG6#\"\"$" }{TEXT 635 13 " . What is " }{XPPEDIT 630 0 "c;" "6#%\"cG" }{TEXT 629 2 " ?" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 329 4 "a) " } {XPPEDIT 452 0 "1/Pi;" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT 445 17 " \+ b) " }{XPPEDIT 453 0 "2/Pi;" "6#*&\"\"#\"\"\"%#PiG!\"\"" } {TEXT 446 16 " c) " }{XPPEDIT 454 0 "3/Pi;" "6#*&\"\"$\"\" \"%#PiG!\"\"" }{TEXT 447 17 " d) " }{XPPEDIT 632 0 "4/Pi; " "6#*&\"\"%\"\"\"%#PiG!\"\"" }{TEXT 631 18 " e) " } {XPPEDIT 634 0 "5/Pi;" "6#*&\"\"&\"\"\"%#PiG!\"\"" }{TEXT 633 8 " \nf ) " }{XPPEDIT 455 0 "Pi;" "6#%#PiG" }{TEXT 448 17 " g) \+ " }{XPPEDIT 456 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 449 17 " \+ h) " }{XPPEDIT 457 0 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 450 17 " i) " }{XPPEDIT 458 0 "Pi/4;" "6#*&%#PiG\" \"\"\"\"%!\"\"" }{TEXT 451 19 " j) " }{XPPEDIT 459 0 "P i/5;" "6#*&%#PiG\"\"\"\"\"&!\"\"" }{TEXT 330 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 511 11 "Solution: c" }}{PARA 0 "" 0 "" {TEXT 512 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> c/(1+x^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&%\"cG\"\"\",&F.F.*$)9$\"\"#F.F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 78 "eqn := Int(f(x), x = 0..sqrt(3)) = 1; \n# A pr obability density integrates to 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$eqnG/-%$IntG6$*&%\"cG\"\"\",&F+F+*$)%\"xG\"\"#F+F+!\"\"/F/;\"\"!*$\" \"$#F+F0F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(va lue, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*(\"\"$!\"\"%# PiG\"\"\"%\"cGF+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c = solve(eqn, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,$*&\"\"$\"\" \"%#PiG!\"\"F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 307 62 "11. A random variable X that assumes values in the interval " } {XPPEDIT 561 1 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT 560 54 " has probab ility density function\n\n " }{XPPEDIT 565 1 "f(x) = \+ 8/((Pi+2)*(1+x^2)^2);" "6#/-%\"fG6#%\"xG*&\"\")\"\"\"*&,&%#PiGF*\"\"#F *F**$,&F*F**$F'F.F*F.F*!\"\"" }{TEXT 562 3 " " }{TEXT 682 7 " for \+ " }{TEXT 563 2 " " }{XPPEDIT 256 1 "0 <= x;" "6#1\"\"!%\"xG" } {XPPEDIT 257 1 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT 564 12 ". \n\nIf \+ " }{XPPEDIT 638 0 "mu[X];" "6#&%#muG6#%\"XG" }{TEXT 637 32 " is the \+ mean of X, what is " }{XPPEDIT 640 0 "(Pi+2)*mu[X];" "6#*&,&%#PiG\" \"\"\"\"#F&F&&%#muG6#%\"XGF&" }{TEXT 639 3 " ? " }}{PARA 0 "" 0 "" {TEXT 306 17 " " }}{PARA 0 "" 0 "" {TEXT 331 5 "a) \+ " }{XPPEDIT 469 0 "2;" "6#\"\"#" }{TEXT 464 16 " b) " } {XPPEDIT 470 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 465 17 " \+ c) " }{XPPEDIT 471 0 "3;" "6#\"\"$" }{TEXT 466 17 " d) " }{XPPEDIT 472 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 467 19 " \+ e) " }{XPPEDIT 473 0 "5;" "6#\"\"&" }{TEXT 468 8 " \nf) \+ " }{XPPEDIT 474 0 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT 463 15 " \+ g) " }{XPPEDIT 475 0 "6;" "6#\"\"'" }{TEXT 462 18 " \+ h) " }{XPPEDIT 476 0 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }{TEXT 461 15 " i) " }{XPPEDIT 477 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrt G6#F$F%" }{TEXT 460 16 " j) " }{XPPEDIT 478 0 "2*sqrt(3); " "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 332 1 "\n" }{TEXT 479 0 " " }}{PARA 0 "" 0 "" {TEXT 509 11 "Solution: a" }}{PARA 0 "" 0 "" {TEXT 510 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eqn := mu[X] = Int(8*x/(1+x^2)^2/(Pi+2), x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/&%#muG6#%\"XG-%$IntG6$,$**\"\")\"\" \"%\"xGF0,&F0F0*$)F1\"\"#F0F0!\"#,&F5F0%#PiGF0!\"\"F0/F1;\"\"!F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(value, eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/&%#muG6#%\"XG,$*&\"\"#\"\"\",& F,F-%#PiGF-!\"\"F-" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 5 "12. " }{TEXT 566 91 "A random real number X is chosen from t he interval [ 1 , 9 ]. If the p.d.f. of X is\n " }{XPPEDIT 684 0 " f(x) = 3*sqrt(x)/52;" "6#/-%\"fG6#%\"xG*(\"\"$\"\"\"-%%sqrtG6#F'F*\"#_ !\"\"" }{TEXT 683 64 ", then what is the probability that X is greate r than 4 ? " }}{PARA 0 "" 0 "" {TEXT 269 4 "a) " }{XPPEDIT 334 1 "7/13;" "6#*&\"\"(\"\"\"\"#8!\"\"" }{TEXT 298 13 " b) " } {XPPEDIT 335 1 "15/26;" "6#*&\"#:\"\"\"\"#E!\"\"" }{TEXT 299 13 " \+ c) " }{XPPEDIT 336 1 "8/13;" "6#*&\"\")\"\"\"\"#8!\"\"" }{TEXT 300 14 " d) " }{XPPEDIT 337 1 "17/26;" "6#*&\"#<\"\"\"\"#E! \"\"" }{TEXT 301 12 " e) " }{XPPEDIT 338 1 "9/13;" "6#*&\"\"* \"\"\"\"#8!\"\"" }{TEXT 302 11 " \n f) " }{XPPEDIT 339 1 "19/26; " "6#*&\"#>\"\"\"\"#E!\"\"" }{TEXT 303 13 " g) " }{XPPEDIT 340 1 "10/13;" "6#*&\"#5\"\"\"\"#8!\"\"" }{TEXT 304 13 " h) \+ " }{XPPEDIT 341 1 "21/26;" "6#*&\"#@\"\"\"\"#E!\"\"" }{TEXT 305 15 " \+ i) " }{XPPEDIT 342 0 "11/13;" "6#*&\"#6\"\"\"\"#8!\"\"" } {TEXT 333 14 " j) " }{XPPEDIT 343 0 "23/26;" "6#*&\"#B\"\"\" \"#E!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 507 11 "Solution: f" }}{PARA 0 "" 0 "" {TEXT 508 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "`P(X>4) =`*Int(3*sqrt(x)/ 52,x = 4 .. 9) = int(3*sqrt(x)/52,x = 4 .. 9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%*P(X>4)~~=G\"\"\"-%$IntG6$,$*(\"\"$F&\"#_!\"\"%\"xG #F&\"\"#F&/F/;\"\"%\"\"*F&#\"#>\"#E" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 588 45 "13. A random variable X has pdf given by " } {XPPEDIT 589 1 "f(x) = (2*x+1)/108;" "6#/-%\"fG6#%\"xG*&,&*&\"\"#\"\" \"F'F,F,F,F,F,\"$3\"!\"\"" }{TEXT 590 8 " for " }{XPPEDIT 591 1 "1 \+ <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 592 1 "`` <= 10;" "6#1%!G\"#5" } {TEXT 593 2 ". " }{TEXT -1 3 " " }{TEXT 594 21 "\nWhat is the median " }{XPPEDIT 595 1 "m;" "6#%\"mG" }{TEXT 596 6 " of " }{XPPEDIT 597 1 "X;" "6#%\"XG" }{TEXT 598 4 "? \n\n" }{TEXT 601 11 "Solution: f " }}{PARA 261 "" 0 "" {TEXT -1 84 "a) 5 b) 16/3 c) 17/3 d) 6 e) 13/2\nf) 7 " }{TEXT 599 66 " \+ g) 22/3 h) 23/3 i) 8 j) 17/2" }} {PARA 3 "" 0 "" {TEXT 600 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x -> (1+2*x)/108;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&#\"\"\"\"$3\"F.*&#F.\"#aF.9$F.F. F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eqn := Int( f(x), x = 1..m)=1/2; # This is setting P(X < m) = 1/2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$,&#\"\"\"\"$3\"F+*&\"#a!\"\"%\"xGF+F+/ F0;F+%\"mG#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn : = map(value, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*&\"$3 \"!\"\"%\"mG\"\"\"F+#F+\"#aF)*&F(F)F*\"\"#F+#F+F/" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "m = solve(eqn,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"mG6$\"\"(!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Of these two values, only 7 is in the i nterval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "By way of verification, we will check that P(X > 7) = 1/2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "testeq( int(f(x), x = 7 .. 1 0) = 1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 308 183 "14. Starting from equilibrium, 60 J of work are expended when a s pring is stretched 2 meters. How many meters beyond equilibrium can t he spring be stretched with a force of 45 N? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "1/ 5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 14 " b) " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 15 " c) " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 14 " d) " }{XPPEDIT 18 0 "6/5;" "6#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT -1 14 " e) " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 20 " \+ \nf) " }{XPPEDIT 18 0 "8/5;" "6#*&\"\")\"\"\"\"\"&!\"\"" } {TEXT -1 15 " g) " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 15 " h) " }{XPPEDIT 18 0 "12/5;" "6#*&\"#7\"\"\"\"\"&!\"\" " }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "5/2;" "6#*&\"\"&\"\"\" \"\"#!\"\"" }{TEXT -1 15 " j) " }{XPPEDIT 18 0 "3;" "6#\"\" $" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 506 12 "Solution: e " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "HookesLaw := F = k*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*HookesLawG/%\"FG*&%\"kG\"\"\"%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn := 60 = Int(k*x, x = 0 .. 2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/\"#g-%$IntG6$*&%\"kG\"\"\"%\"xGF,/F-;\" \"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(valu e, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/\"#g,$*&\"\"#\"\" \"%\"kGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn2 := k = \+ solve(eqn, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"kG\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn3 := subs(\{eqn2, F = \+ 45\}, HookesLaw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/\"#X,$*& \"#I\"\"\"%\"xGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solv e(eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 58 "15. The vertical cross-sections of a tan k have the shape " }{XPPEDIT 686 0 "y = abs(x)-12;" "6#/%\"yG,&-%$abs G6#%\"xG\"\"\"\"#7!\"\"" }{TEXT 685 7 " for " }{XPPEDIT 688 0 "-12 < = x;" "6#1,$\"#7!\"\"%\"xG" }{XPPEDIT 689 0 "`` <= 12;" "6#1%!G\"#7" } {TEXT 687 263 ". Horizontal cross-sections of the tank are rectangles that are 5 m long. The widths of these rectangles vary, but can be d educed from the description of the vertical cross-sections. The tank i s filled to the top with a fluid that has weight density equal to " }{XPPEDIT 569 1 "3/10;" "6#*&\"\"$\"\"\"\"#5!\"\"" }{TEXT 568 1 " " } {XPPEDIT 570 1 "N/(m^3);" "6#*&%\"NG\"\"\"*$%\"mG\"\"$!\"\"" }{TEXT 567 128 ". If after pumping the fluid to the top of the tank, the rem aining fluid is 10 m deep, how many Joules of work have been done?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 3 "a) " } {XPPEDIT 18 0 "24;" "6#\"#C" }{TEXT -1 14 " b) " }{XPPEDIT 18 0 "32;" "6#\"#K" }{TEXT -1 15 " c) " }{XPPEDIT 18 0 "36; " "6#\"#O" }{TEXT -1 10 " d) " }{XPPEDIT 18 0 "48;" "6#\"#[" } {TEXT -1 12 " e) " }{XPPEDIT 18 0 "52;" "6#\"#_" }{TEXT -1 20 " \nf) " }{XPPEDIT 18 0 "56;" "6#\"#c" }{TEXT -1 14 " \+ g) " }{XPPEDIT 18 0 "64;" "6#\"#k" }{TEXT -1 15 " h ) " }{XPPEDIT 18 0 "72;" "6#\"#s" }{TEXT -1 11 " i) " } {XPPEDIT 18 0 "80;" "6#\"#!)" }{TEXT -1 12 " j) " }{XPPEDIT 18 0 "84;" "6#\"#%)" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 572 12 "Solutio n: g " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " `Work =`*Int(3/10*w*5*2*(12-w),w = 0 .. \+ 2) = int(3/10*w*5*2*(12-w),w = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%'Work~=G\"\"\"-%$IntG6$,$*(\"\"$F&%\"wGF&,&\"#7F&F-!\"\"F&F& /F-;\"\"!\"\"#F&\"#k" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 229 " 16. \+ Forty feet of a uniform cable hang over the side of a building. The ca ble weighs 2 lbs/ft. A 48 pound hook is attached to the dangling end o f the cable. The cable is pulled up 30 feet before the hook snags on a balcony rail." }{TEXT -1 1 " " }{TEXT 571 44 "How many foot-pounds of work have been done?" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 371 77 " a) 2480 b) 2540 c) 2580 d) 2640 e) \+ 2680 " }{TEXT 374 1 " " }{TEXT 375 18 " " }}{PARA 0 "" 0 "" {TEXT 372 35 "f) 2740 g) 2800 " }{TEXT -1 1 " " }{TEXT 373 44 "h) 2840 i) 2880 j) 2940 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 504 11 "Solution: j" }}{PARA 0 "" 0 "" {TEXT 505 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "48*30 + (10*2)*30 + int(2* y, y = 0 .. 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%SH" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 19 "17. Calculate " }{XPPEDIT 19 1 " int(1/((8-x)^(1/3)),x = 7 .. 8);" "6#-%$intG6$*&\"\"\"F'),&\"\")F'%\"x G!\"\"*&F'F'\"\"$F,F,/F+;\"\"(F*" }{TEXT 584 1 "." }}{PARA 0 "" 0 "" {TEXT 282 83 "a) 4/3 b) 3/2 c) 5/3 \+ d) 2 e) 7/3" }{TEXT 493 3 " " }{TEXT 285 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 283 36 "f) 5/2 g) \+ 8/3 " }{TEXT -1 5 " " }{TEXT 284 38 "h) 3 \+ i) 10/3 j)" }{TEXT -1 1 " " }{TEXT 585 6 " 7/2 " }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 502 12 "Solution: b " }}{PARA 0 "" 0 "" {TEXT 503 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "I nt(1/(8-x)^(1/3),x = 7 .. 8) = Limit(Int(1/(8-x)^(1/3),x = 7 .. 8-epsi lon), epsilon=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$ *&\"\"\"F(*$),&\"\")F(%\"xG!\"\"#F(\"\"$F(F./F-;\"\"(F,-%&LimitG6%-F%6 $F'/F-;F3,&F,F(%(epsilonGF./F<\"\"!%&rightG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Int(1/(8-x)^(1/3),x = 7 .. 8) = Limit(int(1/(8-x)^ (1/3),x = 7 .. 8-epsilon), epsilon=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$),&\"\")F(%\"xG!\"\"#F(\"\"$F(F./ F-;\"\"(F,-%&LimitG6%,&*(F0F(\"\"#F.%(epsilonG#F9F0F.#F0F9F(/F:\"\"!%& rightG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Int(1/(8-x)^(1/3) ,x = 7 .. 8) = limit(int(1/(8-x)^(1/3),x = 7 .. 8-epsilon), epsilon=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$),& \"\")F(%\"xG!\"\"#F(\"\"$F(F./F-;\"\"(F,#F0\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "18. " }{TEXT 280 12 " Calculate " }{XPPEDIT 256 0 "int(x^2/((2+x^3)^2),x = 1 .. infinity);" "6#-%$intG6$*&%\"xG\"\"#*$ ,&F(\"\"\"*$F'\"\"$F+F(!\"\"/F';F+%)infinityG" }{TEXT 480 1 "." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 481 4 "a) " }{XPPEDIT 490 0 "1; " "6#\"\"\"" }{TEXT 489 15 " b) " }{XPPEDIT 257 0 "1/2;" "6 #*&\"\"\"F$\"\"#!\"\"" }{TEXT 482 17 " c) " }{XPPEDIT 258 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 483 16 " d) \+ " }{XPPEDIT 256 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 487 16 " \+ e) " }{XPPEDIT 259 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT 484 12 " \nf) " }{XPPEDIT 643 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\" " }{TEXT 642 14 " g) " }{XPPEDIT 261 0 "1/7;" "6#*&\"\"\"F$ \"\"(!\"\"" }{TEXT 485 17 " h) " }{XPPEDIT 262 0 "1/8;" " 6#*&\"\"\"F$\"\")!\"\"" }{TEXT 488 17 " i) " }{XPPEDIT 263 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT 486 17 " j) \+ " }{XPPEDIT 264 0 "1/10;" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT 491 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 517 14 "Solution : i \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(x^2/((2+x^3)^2),x = 1 .. infini ty)= Limit(Int(x^2/((2+x^3)^2),x = 1 .. N), N = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"#,&F)\"\"\"*$)F(\"\"$F+ F+!\"#/F(;F+%)infinityG-%&LimitG6$-F%6$F'/F(;F+%\"NG/F:F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "Int(x^2/((2+x^3)^2),x = 1 .. infin ity)= Limit(student[changevar](u = 2+x^3, Int(x^2/((2+x^3)^2),x = 1 .. N), u), N = infinity);\n\n# The change of variable u = 2 + x^3, du = 3*x^2*dx\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"#, &F)\"\"\"*$)F(\"\"$F+F+!\"#/F(;F+%)infinityG-%&LimitG6$-F%6$,$*&F+F+*& F.F+)%\"uGF)F+!\"\"F+/F<;F.,&F)F+*$)%\"NGF.F+F+/FCF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int(x^2/((2+x^3)^2),x = 1 .. infinity)= L imit(int(1/(3*u^2),u = 3 .. 2+N^3), N = infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"#,&F)\"\"\"*$)F(\"\"$F+F+!\"#/F( ;F+%)infinityG-%&LimitG6$,$*(\"\"*!\"\",&F+F9*$)%\"NGF.F+F+F+,&F)F+F;F +F9F+/F=F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int(x^2/((2+x ^3)^2),x = 1 .. infinity)= limit(int(1/(3*u^2),u = 3 .. 2+N^3), N = in finity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"#,&F) \"\"\"*$)F(\"\"$F+F+!\"#/F(;F+%)infinityG#F+\"\"*" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 277 19 "19. Calculate the " }{XPPEDIT 18 0 "100^th; " "6#)\"$+\"%#thG" }{TEXT 690 17 " partial sum of " }{XPPEDIT 19 1 "s um(1/((n+1)*n),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F'*&,&%\"nGF'F 'F'F'F*F'!\"\"/F*;F'%)infinityG" }{TEXT 583 1 "." }}{PARA 260 "" 0 "" {TEXT 492 5 "a) " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT 384 12 " \+ b) " }{XPPEDIT 265 0 "99/100;" "6#*&\"#**\"\"\"\"$+\"!\"\"" } {TEXT 376 13 " c) " }{XPPEDIT 266 0 "100/99;" "6#*&\"$+\"\"\" \"\"#**!\"\"" }{TEXT 377 12 " d) " }{XPPEDIT 256 0 "100/101;" "6#*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT 382 14 " e) " } {XPPEDIT 267 0 "101/100;" "6#*&\"$,\"\"\"\"\"$+\"!\"\"" }{TEXT 378 12 " \nf) " }{XPPEDIT 268 0 "49/50;" "6#*&\"#\\\"\"\"\"#]!\"\"" } {TEXT 379 12 " g) " }{XPPEDIT 273 0 "50/49;" "6#*&\"#]\"\"\"\" #\\!\"\"" }{TEXT 380 13 " h) " }{XPPEDIT 274 0 "98/101;" "6#* &\"#)*\"\"\"\"$,\"!\"\"" }{TEXT 383 14 " i) " }{XPPEDIT 275 0 "101/98;" "6#*&\"$,\"\"\"\"\"#)*!\"\"" }{TEXT 381 13 " j) \+ " }{XPPEDIT 276 0 "1/10100;" "6#*&\"\"\"F$\"&+,\"!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 500 13 "Solution: d " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "answer := sum( 1/((n+1)*n), n = 1.. 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG#\"$+\"\"$,\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "1/1-1/101;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$+\"\"$,\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 279 23 "20. Consider the series" }}{PARA 3 "" 0 "" {TEXT 574 4 "I) " }{XPPEDIT 575 1 "sum(2^n/n,n = 1 .. infinity);" "6#-%$sumG6$*&)\"\"#% \"nG\"\"\"F)!\"\"/F);F*%)infinityG" }{TEXT 576 12 " II) " } {XPPEDIT 577 1 "sum(tan(1/n),n = 1 .. infinity);" "6#-%$sumG6$-%$tanG6 #*&\"\"\"F*%\"nG!\"\"/F+;F*%)infinityG" }{TEXT 578 13 " III) " }{XPPEDIT 579 1 "sum(sec(1/n),n = 1 .. infinity);" "6#-%$sumG6$-%$secG 6#*&\"\"\"F*%\"nG!\"\"/F+;F*%)infinityG" }{TEXT 580 11 " IV) " } {XPPEDIT 581 1 "sum(1/sqrt(n),n = 1 .. infinity);" "6#-%$sumG6$*&\"\" \"F'-%%sqrtG6#%\"nG!\"\"/F+;F'%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 573 72 "List all given series for which t he Divergence Test yields a conclusion." }}{PARA 260 "" 0 "" {TEXT 278 73 "a) I b) II c) III d) IV \+ e) I, II " }}{PARA 260 "" 0 "" {TEXT 582 70 "f) I, III g) I, IV h) II, III i) II, IV j) III, IV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 501 13 "Solution: f " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "a := n -> 2^n/n; \nb := n -> tan(1/n); \nc := n -> \+ sec(1/n);\nd := n -> 1/sqrt(n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)\"\"#9$\"\"\"F/!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"nG6\"6$%)operatorG%&a rrowGF(-%$tanG6#*&\"\"\"F09$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$secG6#*&\"\"\"F09$! \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"nG6\"6$%)op eratorG%&arrowGF(*&\"\"\"F--%%sqrtG6#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "testeq( limit(a(n), n = infinity) = 0 ); #D ivergence Test applies" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "testeq( limit(b(n), n = inf inity) = 0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "testeq( limit(c(n), n = infinity) = 0 ); #Divergence Test applies" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&fa lseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "testeq( limit(d(n), n = infinity) = 0 ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Code for Figures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 692 28 "Code for Problem 1, Figure 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "region := plot([f(x) ,g(x)],x=0..3, color = COLOR(RGB,.05,.65,.99), thickness=2, view=[-5.. 3,0..4]):\nline := plot([-1,y,y=0..4], color=GRAY, linestyle=4):\ntp1 \+ := plots[textplot]([2,1.2,`y = x`]):\ntp2 := plots[textplot]([2.85,4,` y = f(x)`]):\nplots[display](region,line, tp1, tp2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 694 28 "Code for Problem 1, Figure 2" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "surf ace1 := plot3d([(1+y)*cos(t),-1+(1+y)*sin(t),f(y)],t=0..2*Pi,y=0..3,st yle=WIREFRAME):\nsurface2 := plot3d([(1+y)*cos(t),-1+(1+y)*sin(t),g(y) ],t=0..2*Pi,y=0..3,style=WIREFRAME):\nplots[display](surface1,surface2 ,orientation=[15,70],title=\"The Solid of Revolution\",titlefont=[TIME S,BOLD,14]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 695 28 "Code for Problem 1, Figure 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "surface1cut := plot3d([(1 +y)*cos(t),-1+(1+y)*sin(t),f(y)],t=Pi/8..2*Pi-Pi/8,y=0..3,style=WIREFR AME):\nsurface2cut := plot3d([(1+y)*cos(t),-1+(1+y)*sin(t),g(y)],t=Pi/ 8..2*Pi-Pi/8,y=0..3,style=WIREFRAME):\nshell := plot3d([(1+1)*cos(t),- 1+(1+1)*sin(t),z],t=Pi/8..2*Pi-Pi/8,z=g(1)..f(1),style=PATCHNOGRID):\n plots[display](surface1cut,surface2cut,shell,orientation=[-30,75],titl e=\"Cut-away of The Solid of Revolution with One Shell Shown\",titlefo nt=[TIMES,BOLD,14]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 699 28 "Code for Problem 1, F igure 4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "lowerWasher := plot3d([(1+y)*cos(t) ,-1+(1+y)*sin(t),2],t=0..2*Pi,y=2-2^(1/2)..2,style=PATCH):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plots[display](surface1,surface2,l owerWasher,orientation=[20,70],title=\"The Solid of Revolution with a \+ Washer at Height y < 3\", titlefont=[TIMES,BOLD,14]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 701 28 "Code for Problem 1, Figure 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "upperWasher := pl ot3d([(1+y)*cos(t),-1+(1+y)*sin(t),3+1/2],t=0..2*Pi,y=2-1/2*2^(1/2)..2 +1/2*2^(1/2),style=PATCH):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plots[display](surface1,surface2,upperWasher,orientation=[20,70], title=\"The Solid of Revolution with a Washer at Height y > 3\", titl efont=[TIMES,BOLD,14]);" }}}{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 691 28 "Code for Problem 2, Fi gure 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 478 "region := plot([x,x^2],x=0..1, color = COLOR(RGB,.05 ,.65,.99), thickness=2):\nrotatedRegion := plot([4-x,4-x^2],x=0..1, co lor = GRAY, thickness=1):\nline := plot([x,2,x=0..1], color=GRAY, line style=4):\ntp1 := plots[textplot]([1/2,3/4,`y = x`],font=[TIMES, ITALI C, 14]):\ntp2 := plots[textplot]([3/4,1/4,`y = x`],font=[TIMES, ITALIC , 14]):\ntp3 := plots[textplot]([0.81,0.3,`2`], font=[TIMES, ROMAN, 10 ]):\nplots[display](region, rotatedRegion, line, tp1, tp2, tp3, view=[ 0..1,0..4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 704 28 "Code for Problem 2, Figu re 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 761 "paraboloid := plot3d( [(2-x^2)*cos(t), x,2+(2-x^2)*s in(t)], x = 0 .. 1, t = 3*Pi/4 .. 5*Pi/2, style=WIREFRAME, color=SIENN A):\nfrustum := plot3d( [(2-x)*cos(t), x,2+(2-x)*sin(t)], x = 0 .. 1, \+ t = 3*Pi/4 .. 5*Pi/2, style = PATCHNOGRID, color = COLOR(RGB, .99,.9,. 75)):\nwasher1 := plot3d( [r*cos(t), .7,2+r*sin(t)], r = 2-3/4 .. 2-(3 /4)^2, t = 0 .. 2*Pi, style = PATCHNOGRID, color = BROWN):\nwasher2 := plot3d( [r*cos(t), 3/4,2+r*sin(t)], r = 2-3/4 .. 2-(3/4)^2, t = 0 .. \+ 2*Pi, style = PATCHNOGRID, color = TAN):\ncylinder := plot3d( [(2-(3/4 )^2)*cos(t), y,2+(2-(3/4)^2)*sin(t)], y = .7 .. 3/4, t = 0 .. 2*Pi, st yle = PATCHNOGRID, color = COLOR(RGB,.89,.65,.52) ):\nplots[display]( \+ paraboloid, frustum, washer1, washer2, cylinder, orientation = [20,55] ,axes=normal);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 712 28 "Code for Problem 4, Figure 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot3d([x*cos(t),x*sin(t) ,-3/4*(x-2)+3],x=2..6,t=0..2*Pi,style=PATCHNOGRID,orientation=[34,80], scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 713 28 "Code for Problem 7, Figure 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Blue := COLOR(RGB, 0.15,0.75, 0.99): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "fnPl ot := plot(f(x),x=0..2,thickness=2,color=Blue):\nverticalAxis := plot( [x_bar,y,y = 0..4], color = GRAY, linestyle = 4):\nplots[display](fnPl ot, verticalAxis, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 714 28 "Code for Problem 8, Figure 1" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "horizontalAxis := plot(y_bar, x=0 .. 2, color = G RAY, linestyle = 4):\ndot := plot([x_bar+0.03*cos(t),y_bar+0.03*sin(t) ,t=0..2*Pi], color=COLOR(RGB,.3,.3,.3)):\nplots[display](fnPlot, verti calAxis, horizontalAxis, dot, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1 14 0" 68 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }