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0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Volumes of solids of revolution . . II" }}{PARA 3 "" 0 "" {TEXT 263 57 " .. Volumes of revolution for re gions rotated around the " }{TEXT 265 1 "y" }{TEXT 266 6 " axis " }} {PARA 0 "" 0 "" {TEXT -1 63 "by Peter Stone, Mathematics Dept., Malasp ina University-College" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 3.3.20 05 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load calculus procedures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "read \"E:\\\\MapleMath/procdrs/calculus.m \";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The shell method for determining volumes of revolution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "S uppose that the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 16 " lies above the " }{TEXT 290 1 "x" }{TEXT -1 70 " axis, o r at least does not go below it, throughout the interval from " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 8 ", where " }{TEXT 279 1 "a" }{TEXT -1 35 " is greater than or equal to 0 and " }{XPPEDIT 18 0 "a0" "6#0%\"aG\"\"!" }{TEXT -1 56 ", the solid of revolution will have a \"hole\" through it." }}{PARA 0 "" 0 "" {TEXT -1 28 "Subdivide the interval from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\" aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 52 " into a large number of subintervals of equal width " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 268 1 "x" }{TEXT -1 81 ", and conside r the corresponding subdivision of the region between the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 9 " and the \+ " }{TEXT 293 1 "x" }{TEXT -1 36 " axis into vertical strips of width \+ " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 267 1 "x" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 638 408 408 {PLOTDATA 2 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[l-%%FONTG6$%*HELVETICAGFhgr-F_ir6&7$$\"$2\"F`bm$\"#DFa[lQ\"xFeirF[[lF fir-F_ir6&7$$\"+++++]Fh[mFh[lFajrF[[lFfir-F_ir6&7$$\"$.\"F`bmF_jrQ\"dF eirF[[l-Fgir6$Fi\\qFhgr-%+AXESLABELSG6%Q!FeirFb[s-Fgir6#%(DEFAULTG-%*A XESSTYLEG6#%%NONEG-%%VIEWG6$Fe[sFe[s" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" }}{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "This " }{TEXT 259 14 "volume element" }{TEXT -1 72 " may be thought of as the \"thickened\" surface of a cyl inder with radius " }{TEXT 281 1 "x" }{TEXT -1 12 " and height " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 131 "Imagine that this cylindrical shell is sliced open \+ with a vertical cut, and opened out to form a (thickened) rectangle wi th length " }{XPPEDIT 18 0 "2*Pi*x;" "6#*(\"\"#\"\"\"%#PiGF%%\"xGF%" } {TEXT -1 9 ", height " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 15 " and thickness " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" } {TEXT 282 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 823 306 306 {PLOTDATA 2 "6<-%'CURVESG6$7'7$$\"\"!F)F(7$F($\" \"#F)7$$\"3;+++++++s!#$\"3%)************p?F07$$\"3e************p sF0FBF-F2-F$6$7%FD7$FEF?F1F2-%)POLYGONSG6%7&F'F*7$$\"#sF8F+7$FPF(-F36& F5$\"#&*!\"#$\"#\"*FW\"\"\"-%&STYLEG6#%,PATCHNOGRIDG-FL6%7&F*7$$F:FW$ \"$2#FW7$$\"$F(FWF^oFO-F36&F5$\"$`*!\"$$\"$B*FgoFZFen-FL6%7&FOF`o7$Fao F]oFR-F36&F5FU$\"#*)FWFZFen-F$6%7$F'7$F($!3))**************H!#=F2-%*LI NESTYLEG6#F,-F$6%7$F17$F.FfpF2Fip-F$6%7$F'7$$!3++++++++]FhpF(F2Fip-F$6 %7$F*7$FdqF+F2Fip-F$6%7$F-7$$\"3M+++++++zF0F+F2Fip-F$6%7$FD7$$\"3k**** *********4)F0FBF2Fip-F$6&7$7$$\"+++++L!\"*$FWF87$F(F]s7%7$$\")+++**F\\ s$!+++++C!#5F^s7$Fas$!+++++;FesFenF2-F$6&7$7$$\"+++++RF\\sF]s7$$\"++++ +sF\\sF]s7%7$$\"++++,rF\\sFgsF_t7$FdtFcsFenF2-F$6&7$7$$FgoF8$\"$8\"FW7 $F[u$\"$+#FW7%7$$!+++++NFes$\"+++5R>F\\sF^u7$$!+++++DFesFeuFenF2-F$6&7 $7$F[u$\"#()FW7$F[uF(7%7$Fhu$\"*+++4'FesF`v7$FcuFcvFenF2-F$6&7$7$$\"#v F8$\"#=F87$$\"#xF8$\"#?F87%7$$\"+j7D:wF\\s$\"+P([Z'>F\\sF^w7$$\"+P([Zm (F\\s$\"+j7D:>F\\sFenF2-F$6&7$7$$\"$(zFW$\"$F#FW7$$\"$x(FWF^o7%7$$\"+P ([Z&yF\\s$\"+j7D0@F\\sFfx7$$\"+j7D0yF\\s$\"+P([Z:#F\\sFenF2-%%TEXTG6&7 $$\"+++++OF\\sF]sQ'2~~~~x6\"F2-%%FONTG6$%*HELVETICAG\"#5-Fey6&7$F_w$\" #AF8Q\"xF[zF2F\\z-Fey6&7$F[u$FZF)Q%f(x)F[zF2F\\z-Fey6&7$$\"++++5OF\\sF ]sQ\"pF[zF2-F]z6$%'SYMBOLGF`z-Fey6&7$$\"$i(FWFdzQ\"dF[zF2Fb[l-%+AXESLA BELSG6%Q!F[zF^\\l-F]z6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$Fa\\l Fa\\l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+ 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "C urve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "This indicates that we should take the volume of the volume ele ment to be: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi* x*f(x)*delta;" "6#*,\"\"#\"\"\"%#PiGF%%\"xGF%-%\"fG6#F'F%%&deltaGF%" } {TEXT 283 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "Another way to see this is as follows. If " }{TEXT 287 1 "x" }{TEXT -1 26 " is the distance from the " }{TEXT 288 1 "y" }{TEXT -1 58 " axis to the mid-point of the subinterval of t he interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 141 " \+ which determines the cylindrical shell, the volume of the cylindrical \+ shell is obtained by subtracting the volume of a cylinder with height \+ " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " and radius \+ " }{XPPEDIT 18 0 "x-delta*x/2" "6#,&%\"xG\"\"\"*(%&deltaGF%F$F%\"\"#! \"\"F)" }{TEXT -1 43 " from the volume of a cylinder with height " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " and radius " } {XPPEDIT 18 0 "x+delta*x/2" "6#,&%\"xG\"\"\"*(%&deltaGF%F$F%\"\"#!\"\" F%" }{TEXT -1 14 ". This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Pi*(x+delta*x/2)^2*f(x)-Pi*(x-delta*x/2)^2*f(x);" "6#,& *(%#PiG\"\"\"*$,&%\"xGF&*(%&deltaGF&F)F&\"\"#!\"\"F&F,F&-%\"fG6#F)F&F& *(F%F&*$,&F)F&*(F+F&F)F&F,F-F-F,F&-%\"fG6#F)F&F-" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*((x+delta*x/2 )^2-(x+delta*x/2)^2)*f(x);" "6#/%!G*(%#PiG\"\"\",&*$,&%\"xGF'*(%&delta GF'F+F'\"\"#!\"\"F'F.F'*$,&F+F'*(F-F'F+F'F.F/F'F.F/F'-%\"fG6#F+F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*((x^2+x*delta*x+delta*x^2/4)^2-(x^2-x*delta*x+delta*x^2/4))*f(x); " "6#/%!G*(%#PiG\"\"\",&*$,(*$%\"xG\"\"#F'*(F,F'%&deltaGF'F,F'F'*(F/F' *$F,F-F'\"\"%!\"\"F'F-F',(*$F,F-F'*(F,F'F/F'F,F'F3*(F/F'*$F,F-F'F2F3F' F3F'-%\"fG6#F,F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*``(2*x*delta* x)*f(x);" "6#/%!G*(%#PiG\"\"\"-F$6#**\"\"#F'%\"xGF'%&deltaGF'F,F'F'-% \"fG6#F,F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*x*f(x)*delta;" "6 #/%!G*,\"\"#\"\"\"%#PiGF'%\"xGF'-%\"fG6#F)F'%&deltaGF'" }{TEXT 289 1 " x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 124 "An estimate of the total volume of the solid of revolu tion can be obtained by adding the volumes of such volume elements as \+ " }{TEXT 274 1 "x" }{TEXT -1 13 " varies from " }{TEXT 275 1 "a" } {TEXT -1 4 " to " }{TEXT 276 1 "b" }{TEXT -1 64 ", that is, the volume can be approximated by a sum of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(2*Pi*x*f(x)*delta,x = a .. b);" "6# -%$SumG6$*,\"\"#\"\"\"%#PiGF(%\"xGF(-%\"fG6#F*F(%&deltaGF(/F*;%\"aG%\" bG" }{TEXT 271 1 "x" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{TEXT 272 1 "x" }{TEXT -1 94 " runs over a finite sequence of \+ values giving the locations of vertical strips of equal width " } {XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 273 1 "x" }{TEXT -1 61 " whi ch cover the whole of the region between the graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6# /%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 102 "Now cons ider taking progressively narrower strips, and correspondingly thinner cylindrical shells. As " }{XPPEDIT 18 0 "delta" "6#%&deltaG" } {XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F *" }{TEXT -1 83 " and the number of strips tends to infinity, the sum \+ above approaches the integral " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(2*Pi*x*f(x),x = a .. b);" "6#-%$IntG6$**\"\"#\"\"\" %#PiGF(%\"xGF(-%\"fG6#F*F(/F*;%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 87 "The volume of the solid of revolution formed by ro tating the region between the graphs " }{XPPEDIT 18 0 "y=f(x)" "6#/%\" yG-%\"fG6#%\"xG" }{TEXT -1 9 " and the " }{TEXT 294 1 "x" }{TEXT -1 11 " axis from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 12 " around the " } {TEXT 277 1 "y" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*f(x),x = a .. b);" "6#/&%\" VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*-%\"fG6#F0F*/F0;%\"aG% \"bGF*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 111 "It is sometimes possible to calculate v olumes of solids of revolution obtained by rotating a region around th e " }{TEXT 284 1 "y" }{TEXT -1 44 " axis by using horizontal discs or \+ washers. " }}{PARA 0 "" 0 "" {TEXT -1 102 "For example, the volume of \+ the solid of revolution obtained by rotating the region bounded by a c urve " }{XPPEDIT 18 0 "x=g(y)" "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 6 ", \+ the " }{TEXT 285 1 "y" }{TEXT -1 20 " axis and the lines " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=d" "6 #/%\"xG%\"dG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "c " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl e 1 " }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 110 "Find the volume of the solid of revoluti on formed when the region in the first quadrant bounded by the curve \+ " }{XPPEDIT 18 0 "y = 4-x^2;" "6#/%\"yG,&\"\"%\"\"\"*$%\"xG\"\"#!\"\" " }{TEXT -1 23 " is rotated around the " }{TEXT 295 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 8 "Method I" }{TEXT -1 11 ": - using " }{TEXT 259 18 "cylindrical shells" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 670 "f := x -> 4-x^2: \np1 := plot(f(x),x=0..2.3,y=0..4.2,color=red,\n thickness=2,linesty le=1):\np2 := plot(f(-x),x=-2.3..0,y=0..4.2,color=red,\n thickness=1 ,linestyle=2):\na := 0: b := 2: c := 1.1:\np3 := plot(f(x),x=a..b,colo r=COLOR(RGB,.9,.9,.9),filled=true):\nh := evalf(Pi/20): r := .2:\np4 : = plot([seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i=0..40)],\n linestyl e=2,color=black):\np5 := plot([[seq([c*cos(i*h),f(c)+r*c*sin(i*h)],i=0 ..40)],\n [seq([c*cos(i*h),r*c*sin(i*h)],i=0..40)],\n \+ [[c,0],[c,f(c)]],[[-c,0],[-c,f(c)]]],\n linestyle=1,color=CO LOR(RGB,.7,.7,.7),thickness=2):\nplots[display]([p1,p2,p3,p4,p5],tickm arks=[3,5],view=[-2.3..2.3,-.4..4.2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 363 280 280 {PLOTDATA 2 "6--%'CURVESG6&7S7$$\"\"!F)$\"\"%F)7$$\"3qKLLe aM8]!#>$\"3qS:tOm[(*R!#<7$$\"3WmmTN0Vv$*F/$\"3ydvAI,@\"*RF27$$\"3ALL3U ^5G9!#=$\"3g^:.d^gzRF27$$\"3VLLe%**=>#>F;$\"3!yf#ytA1jRF27$$\"3Hm;/OeQ 8CF;$\"3%[$R1)ob<%RF27$$\"3AL$3-^Q!pGF;$\"3i:%p-='oYz8QF27$$\"3_mm;**pW:[F;$\"3En@)>q9\"oPF27$$\"3iKLezp 5c_F;$\"3k`<>%RLPs$F27$$\"3W++]Zd=_dF;$\"3OyDE\"fB\"pOF27$$\"3u****\\i 9I]iF;$\"3p.(yiJP$4OF27$$\"3m****\\_#G.t'F;$\"3-.g8;o-ZNF27$$\"3oK$3_c Qi;(F;$\"3+3_E[-X'[$F27$$\"3'fmm\"*3yXo(F;$\"32'*\\\"ffs%4MF27$$\"3\"e mmmlzO7)F;$\"3&=ge$)Ge+M$F27$$\"3E**\\(o;fWj)F;$\"3'Hob*[6YaKF27$$\"3? lmm\"e&e'3*F;$\"3qu/nC'RV<$F27$$\"3#)**\\(o\"*REe*F;$\"3wQ^BA,t\"3$F27 $$\"3$**\\i]4+b+\"F2$\"3s.$H)e&p*))HF27$$\"3om;a8gya5F2$\"3))yojlkU()G F27$$\"3gmT5se/+6F2$\"3k$em#z!**)*y#F27$$\"3GL$eRuk)[6F2$\"3c`&*3+)4,o #F27$$\"3GL3_JQd*>\"F2$\"3eF/tAE-hDF27$$\"3\")*\\78C;PC\"F2$\"3'y]&46* pJX#F27$$\"3OL$3KD\"R\"H\"F2$\"3[5q8J'3BL#F27$$\"3&)****\\0UkS8F2$\"3# RFfU6tE?#F27$$\"3#)**\\P5'G))Q\"F2$\"3?ZQ+\"4b62#F27$$\"3!)*\\(o*\\\\a V\"F2$\"3c1p%Qt%[R>F27$$\"3%)**\\i[S@([\"F2$\"3#oVsNP%>)y\"F27$$\"3Ymm ;)zEP`\"F2$\"3kDae3@oZ;F27$$\"3'****\\i3*Q$e\"F2$\"3C!RW:+zG\\\"F27$$ \"3,L3_+0RG;F2$\"376N\"yPW$[8F27$$\"3)****\\F$*)ex;F2$\"3#fr9FP&p&=\"F 27$$\"3[mTNB3)Qs\"F2$\"36h<(o!\\BG5F27$$\"3%)*\\Pu=pAx\"F2$\"3oUifs#>1 f)F;7$$\"3[mm\"zlx&>=F2$\"3M&)\\5o9P\"*oF;7$$\"3%)*\\(=U_5p=F2$\"3Yw?F OfXk]F;7$$\"3FLLL#>1o\">F2$\"3[')eK5-aeKF;7$$\"35L$eMI(el>F2$\"3i&)R$[ _lYO\"F;7$$\"3ZmTN#[kR,#F2$!3>-'3%=OH0cF/7$$\"31++]&3=%e?F2$!3)3&[8Z,& 3P#F;7$$\"3_m;HJpO4@F2$!3!\\Ao#3&)G%\\%F;7$$\"3'HLLj=O\\:#F2$!3\"p[)or '*\\PkF;7$$\"3))*\\(o;D_.AF2$!3!=8)[:[6b&)F;7$$\"3#**\\7V$e-]AF2$!33hO 2bihi5F27$$\"3#)*************H#F2$!39*************G\"F2-%'COLOURG6&%$R GBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#\"\"\"-F$6&7S7 $$!3#)*************H#F2Fgz7$$!3UmmTXl')\\AF2$!3EC^=s%**=1\"F27$$!3JLek %pXi?#F2$!35bFV^1?v')F;7$$!3qm;z&[*=d@F2$!3\"y:,hxkY`'F;7$$!3Xm;a+\"3y 5#F2$!3J!o!4())\\&GWF;7$$!3ALeR;9me?F2$!3o&)*oNFo3Q#F;7$$!3Lm\"z*[h48? 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1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "f(x) = 4-x^2;" "6#/-%\"fG6#%\"xG,&\"\"%\"\"\"*$F'\" \"#!\"\"" }{TEXT -1 32 ". Then the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*f(x),x = 0 .. 2);" "6#/&%\"VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*-%\"fG6#F 0F*/F0;\"\"!F)F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2*Pi*Int(x*(4-x^2),x = 0 .. 2);" "6#/%!G*(\"\"#\" \"\"%#PiGF'-%$IntG6$*&%\"xGF',&\"\"%F'*$F-F&!\"\"F'/F-;\"\"!F&F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(4*x-x^3,x = 0 .. 2);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$,& *&\"\"%F'%\"xGF'F'*$F/\"\"$!\"\"/F/;\"\"!F&F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*Pi*(2*x^2-x^4/4) " "6#/%!G*(\"\"#\"\"\"%#PiGF',&*&F&F'*$%\"xGF&F'F'*&F,\"\"%F.!\"\"F/F' " }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2,``],[0,``])" "6#-%*PIEC EWISEG6$7$\"\"#%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*( 4-0);" "6#/%!G*(\"\"#\"\"\"%#PiGF',&\"\"%F'\"\"!!\"\"F'" }{XPPEDIT 18 0 " ``=8*Pi" "6#/%!G*&\"\")\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 299 1 "~" }{TEXT -1 14 " 25.13274123. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f := x -> 4-x^2;\n2*Pi*Int(x *f(x),x=0..2);\nvalue(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"%\"\"\"* $)9$\"\"#F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\" \"\"%#PiGF&-%$IntG6$*&%\"xGF&,&\"\"%F&*$)F,F%F&!\"\"F&/F,;\"\"!F%F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\")\"\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BTF8D!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 9 "Method II" }{TEXT -1 11 ": - usin g " }{TEXT 259 16 "horizontal discs" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " } {XPPEDIT 18 0 "y=4-x^2" "6#/%\"yG,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" } {TEXT -1 19 " can be solved for " }{TEXT 302 1 "x" }{TEXT -1 13 " in t erms of " }{TEXT 301 1 "y" }{TEXT -1 9 " to give " }{XPPEDIT 18 0 "x=s qrt(4-y)" "6#/%\"xG-%%sqrtG6#,&\"\"%\"\"\"%\"yG!\"\"" }{TEXT -1 7 ", w hen " }{XPPEDIT 18 0 "x>=0" "6#1\"\"!%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(y)=sqrt(4-y)" "6#/-%\"gG 6#%\"yG-%%sqrtG6#,&\"\"%\"\"\"F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "Then the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y]=Pi*Int(g(y)^2,y=0..4)" "6#/&%\"VG6 #%\"yG*&%#PiG\"\"\"-%$IntG6$*$-%\"gG6#F'\"\"#/F';\"\"!\"\"%F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*Int( 4-y,y=0..4)" "6#/%!G*&%#PiG\"\"\"-%$IntG6$,&\"\"%F'%\"yG!\"\"/F-;\"\"! F,F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*(4*y-y^2/2)" "6#/%!G*&%#PiG\"\"\",&*&\"\"%F'%\"yGF'F'*&F+\"\" #F-!\"\"F.F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4,``],[0,``]) " "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=8*Pi" "6#/%!G*&\"\")\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 303 1 "~ " }{TEXT -1 14 " 25.13274123. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "g := y -> sqrt(4-y);\nPi*Int (g(y)^2,y=0..4);\nvalue(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"yG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\" \"%\"\"\"9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\" \"-%$IntG6$,&\"\"%F%%\"yG!\"\"/F+;\"\"!F*F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\")\"\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BTF8D!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "revsolid(4-x^2,x=0..2,axis=YAXIS,scaling= constrained,tickmarks=[0,5,5],orientation=[41,47]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 310 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 110 "Find the volume of the solid of revolution formed when the region in the first quadrant bounded by the curve " }{XPPEDIT 18 0 "y = exp (x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 14 " and the line " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 12 " is rotated " }} {PARA 0 "" 0 "" {TEXT -1 15 "(a) around the " }{TEXT 312 1 "x" }{TEXT -1 23 " axis, (b) around the " }{TEXT 309 1 "y" }{TEXT -1 7 " axis. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "f := x -> exp(x):\np1 := plot([f(x),-f(x)],x=0..1.4,y=-3..3,color=red,\n \+ thickness=[2,1],linestyle=[1,2]):\na := 0: b := 1:\np2 := plot(f(x),x =a..b,color=COLOR(RGB,.85,.85,.85),filled=true):\np3 := plot([[[a,0],[ a,f(a)]],[[b,0],[b,f(b)]]],color=black):\nh := evalf(Pi/20): r := .08: \np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)],\n \+ [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n linestyle =2,color=black):\nplots[display]([p1,p2,p3,p4],tickmarks=[2,7],view=[- .2..1.4,-3..3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 243 469 469 {PLOTDATA 2 "6,-%'CURVESG6&7S7$$\"\"!F)$\"\"\"F)7$$\"3+LLL$e,;0$!#>$\"3zPH--k)4. \"!#<7$$\"3fmm;/Qy1dF/$\"3Y)\\o*Hwse5F27$$\"3!GLL$3R\"Gp)F/$\"3c2&*p*G =34\"F27$$\"3KLL$etj)p6!#=$\"39n^27T5C6F27$$\"3cmmTlu,p9F@$\"3cb!Qj:S# e6F27$$\"37LL3n7PY\"F27$$\"3n***\\P1bN.#F@$\"3_JUum! 3bA\"F27$$\"3+LL3d4cIBF@$\"33an\\&H_CE\"F27$$\"3))***\\([VhEEF@$\"3H6r $*yjQ+8F27$$\"3jmmm^;9JHF@$\"3QX?/JefS8F27$$\"3ELL$eYp$*>$F@$\"3#zriW$ 4/x8F27$$\"3i*****\\XI8]$F@$\"3M#)[qhjD>9F27$$\"3e*****\\KJX!QF@$\"39' Q`#[t%HY\"F27$$\"3'******\\a@n4%F@$\"3E>z\"F27$$\"3a***\\7ON /7'F@$\"3Q9rIKi>W=F27$$\"31mm;/mV?kF@$\"3>J)z_gg.!>F27$$\"3gmmT&RJfp'F @$\"3c*zp/QUM&>F27$$\"3)HLLeu*3$*pF@$\"3A+d&QjhB,#F27$$\"3TKL3dPv,tF@$ \"3Eg'=$eXWv?F27$$\"3i)**\\ioY/d(F@$\"3O9mLMi'>8#F27$$\"3#GLL3TU1'yF@$ \"3I(\\\"GK9u%>#F27$$\"3i)******)HWg\")F@$\"3#\\&ybfh`hAF27$$\"3K)*** \\n$RPX)F@$\"3\\Bw!Q\\[)GBF27$$\"3M***\\Pp=vt)F@$\"3s;+*>0$)eR#F27$$\" 3)*)***\\_sg_!*F@$\"3'\\Y^@]wDZ#F27$$\"3NkmmO$GdL*F@$\"3eO;I`2eVDF27$$ \"3]*****\\_?!Q'*F@$\"3f=%*H\"=X;i#F27$$\"3eJL3x@%>\"**F@$\"35zv.7.X%p #F27$$\"3$*****\\*3T6-\"F2$\"3UOeaL5OwFF27$$\"3Um;/i(=$\\5F2$\"3!p/'fS ]qbGF27$$\"3'***\\()[Dxy5F2$\"3yq4@=t1THF27$$\"3cmm;4!pv5\"F2$\"3I(\\J e&3*p-$F27$$\"3!)**\\PMirP6F2$\"3jkk$oqN'>JF27$$\"3QLLL&f^n;\"F2$\"3aw 1m$GV:@$F27$$\"3=LLeXWW'>\"F2$\"3L@?!z-L$3LF27$$\"3^m;/C9*eA\"F2$\"3s( o=7*>?2MF27$$\"3'*******Q,&HD\"F2$\"3*HVx!e^l+NF27$$\"3omm\"*zC'RG\"F2 $\"3b.bW6'>4h$F27$$\"3ALLL(G+qLRF27$$\"3!******* *******R\"F2$\"3auY%o'**>bSF2-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#F+-%'C OLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7S7$F($!\"\"F)7$F-$!3zPH--k)4.\"F 27$F4$!3Y)\\o*Hwse5F27$F9$!3c2&*p*G=34\"F27$F>$!39n^27T5C6F27$FD$!3cb! 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" }}{PARA 0 "" 0 "" {TEXT -1 46 "Using v ertical discs the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int(f(x)^2,x = 0 .. 1);" "6#/&%\"VG6#% \"xG*&%#PiG\"\"\"-%$IntG6$*$-%\"fG6#F'\"\"#/F';\"\"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(exp (x)^2,x = 0 .. 1);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$*$-%$expG6#%\"xG\"\"# /F/;\"\"!F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Pi*Int(exp(2*x),x = 0 .. 1);" "6#/%!G*&%#PiG\"\"\" -%$IntG6$-%$expG6#*&\"\"#F'%\"xGF'/F0;\"\"!F'F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi;" "6#/%!G%#Pi G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*x)/2;" "6#*&-%$expG6#*&\"\"# \"\"\"%\"xGF)F)F(!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(exp(2)/2 -1/2);" "6#/%!G*&%#PiG\"\"\",&*&-%$expG6#\"\"#F'F-!\"\"F'*&F'F'F-F.F.F '" }{XPPEDIT 18 0 "`` = Pi/2;" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp(2)-1)" "6#-%!G6#,&-%$expG6#\"\"#\"\"\" F+!\"\"" }{TEXT -1 1 " " }{TEXT 313 1 "~" }{TEXT -1 14 " 10.03590585. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x -> exp(x);\nPi*Int(f(x)^2,x=0..1);\nvalue(%);\nevalf(ev alf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-%$IntG6$*$)-%$expG6#%\"xG\"\"# F%/F.;\"\"!F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\",&*&#F %\"\"#F%*$)-%$expG6#F%F)F%F%F%#F%F)!\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&e!f.5!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 776 "f := x -> exp(x):\np1 := pl ot(f(x),x=0..1.4,y=0..3,color=red,\n thickness=2,linestyle=1):\np2 : = plot(f(-x),x=-1.4..0,y=0..3,color=red,\n thickness=1,linestyle=2): \na := 0: b := 1: c := .6:\np3 := plot(f(x),x=a..b,color=COLOR(RGB,.9, .9,.9),filled=true):\nh := evalf(Pi/20): r := .2:\np4 := plot([[seq([b *cos(i*h),f(b)+r*b*sin(i*h)],i=0..40)],\n [seq([b*cos(i*h),r*b*sin (i*h)],i=0..40)],\n [[-b,0],[-b,f(b)]]],linestyle=2,color=black): \np5 := plot([[b,0],[b,f(b)]],color=black):\np6 := plot([[seq([c*cos(i *h),f(c)+r*c*sin(i*h)],i=0..40)],\n [seq([c*cos(i*h),r*c*sin (i*h)],i=0..40)],\n [[c,0],[c,f(c)]],[[-c,0],[-c,f(c)]]],\n \+ linestyle=1,color=COLOR(RGB,.7,.7,.7),thickness=2):\nplots[displ ay]([p1,p2,p3,p4,p5,p6],tickmarks=[3,5],view=[-1.4..1.4,-.2..3]);" }} {PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "60-%'CURVESG6&7S7$ $\"\"!F)$\"\"\"F)7$$\"3+LLL$e,;0$!#>$\"3zPH--k)4.\"!#<7$$\"3fmm;/Qy1dF /$\"3Y)\\o*Hwse5F27$$\"3!GLL$3R\"Gp)F/$\"3c2&*p*G=34\"F27$$\"3KLL$etj) 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p$!3********>R?3PF/7$F`eo$!35+++uN@x=F/7$F[eo$\"3o*****fFi\\%)*Fd`oFj^ pF`[lFd[l-F$6&7$Fa_pFjdoFj^pF`[lFd[l-F$6&7$7$F]joF(F\\joFj^pF`[lFd[l-% *AXESTICKSG6$\"\"$\"\"&-%+AXESLABELSG6%Q\"x6\"Q\"yF`gp-%%FONTG6#%(DEFA ULTG-%%VIEWG6$;$!#9Ff`n$\"#9Ff`n;$!\"#Ff`n$FjfpF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "revsolid(exp(x),x=0..1,axis=YAXIS,scaling=constrained ,tickmarks=[0,3,3],orientation=[33,44]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Using cylindrical shells \+ the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*f(x),x = 0 .. 1);" "6#/&%\"VG6#%\"yG* (\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*-%\"fG6#F0F*/F0;\"\"!F*F*" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*exp(x),x = 0 .. 1);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$* &%\"xGF'-%$expG6#F-F'/F-;\"\"!F'F'" }{TEXT -1 10 " ... " } {XPPEDIT 18 0 "PIECEWISE([u=x,v=exp(x)],[du/dx=1,dv/dx=exp(x)])" "6#-% *PIECEWISEG6$7$/%\"uG%\"xG/%\"vG-%$expG6#F)7$/*&%#duG\"\"\"%#dxG!\"\"F 3/*&%#dvGF3F4F5-F-6#F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*Pi" "6#/%!G *&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 314 1 "[" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x*exp(x)" "6#*&%\"xG\"\"\"-%$expG6#F$F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``])" "6#-%*PIECEWISEG6$7$ \"\"\"%!G7$\"\"!F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-Int(exp(x),x=0.. 1)" "6#,$-%$IntG6$-%$expG6#%\"xG/F*;\"\"!\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 315 1 "]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "(using the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x )=u*v-Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#d xG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =2*Pi" "6#/%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 327 1 "[" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(x)" "6#*&%\"xG\"\"\"-%$expG6#F$F% " }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``])" "6#-%*PI ECEWISEG6$7$\"\"\"%!G7$\"\"!F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``-ex p(x)" "6#,&%!G\"\"\"-%$expG6#%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F( " }{TEXT -1 1 " " }{TEXT 326 1 "]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 2*Pi*(exp(1)-(exp(1)-1));" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-%$expG6#F'F', &-F+6#F'F'F'!\"\"F0F'" }{XPPEDIT 18 0 "``=2*Pi" "6#/%!G*&\"\"#\"\"\"%# PiGF'" }{TEXT -1 1 " " }{TEXT 316 1 "~" }{TEXT -1 13 " 6.283185307." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> exp(x);\n2*Pi*Int(x*f(x),x=0..1);\nvalue(%);\nevalf(ev alf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&-%$e xpG6#F,F&/F,;\"\"!F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"# \"\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2`=$G'!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Alternatively, the equation " } {XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 19 " ca n be solved for " }{TEXT 351 1 "x" }{TEXT -1 13 " in terms of " } {TEXT 350 1 "y" }{TEXT -1 9 " to give " }{XPPEDIT 18 0 "x = ln*y;" "6# /%\"xG*&%#lnG\"\"\"%\"yGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 73 "Then the required volume can be obtained by first calculating t he volume " }{XPPEDIT 18 0 "V[1]" "6#&%\"VG6#\"\"\"" }{TEXT -1 103 " o f the solid of revolution obtained by rotating the region in the first quadrant bounded by the curve " }{XPPEDIT 18 0 "x = ln*y;" "6#/%\"xG* &%#lnG\"\"\"%\"yGF'" }{TEXT -1 11 ", the line " }{XPPEDIT 18 0 "y=exp( 1)" "6#/%\"yG-%$expG6#\"\"\"" }{TEXT -1 9 " and the " }{TEXT 352 1 "y " }{TEXT -1 48 " axis (the region ABD in the picture) about the " } {TEXT 353 1 "y" }{TEXT -1 28 " axis, and then subtracting " }{XPPEDIT 18 0 "V[1]" "6#&%\"VG6#\"\"\"" }{TEXT -1 17 " from the volume " } {XPPEDIT 18 0 "V[2]" "6#&%\"VG6#\"\"#" }{TEXT -1 51 " obtained by rota ting the rectangle ABCD about the " }{TEXT 354 1 "y" }{TEXT -1 25 " ax is. The latter volume " }{XPPEDIT 18 0 "V[2]" "6#&%\"VG6#\"\"#" } {TEXT -1 48 " is that of a cylinder with radius 1 and height " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 5 ", so " } {XPPEDIT 18 0 "V[2]=Pi*exp(1)" "6#/&%\"VG6#\"\"#*&%#PiG\"\"\"-%$expG6# F*F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 190 231 231 {PLOTDATA 2 "6--%'CURVESG6%7S7$$\"\"!F)$\"\"\"F)7$$\"3WLLL epo(R#!#>$\"3$[y-fimU-\"!#<7$$\"35nmTg:!R[%F/$\"3k`$*Hz%fe/\"F27$$\"3! 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\"3*>O4Z;k?]#F27$Fg^mF)7&F[_mFf^m7$$\"3ImmmmxGp$*FE$\"3e[Xr/78_DF27$F^ _mF)7&Fb_mF]_m7$$\"3A++D\"oK0e*FE$\"335')HYrh1EF27$Fe_mF)7&Fi_mFd_m7$$ \"3A++v=5s#y*FE$\"3!*=xB3j&)fEF27$F\\`mF)7&F``mF[`m7$F*$\"34X!f%G=G=FF 27$F*F)7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&F\\[l$\"#&)!\"#F^amF^am- F$6%7%7$F*F(Fb`m7$F(Fc`m-%*LINESTYLEG6#F+-Fjz6&F\\[lF)F)F)-%%TEXTG6%7$ $F_[lF`am$\"#F!\"\"Q\"A6\"Fiam-F\\bm6%7$$\"$3\"F`amF`bmQ\"BFdbmFiam-F \\bm6%7$Fhbm$!#8F`amQ\"CFdbmFiam-F\\bm6%7$$!\"(F`amF^cmQ\"OFdbmFiam-F \\bm6%7$$\"\"(F`am$\"$D\"F`amQ\"DFdbmFiam-%+AXESLABELSG6%Q\"xFdbmQ\"yF dbm-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$7#/F+%\"1G7%F\\em/Fc[l%\"2G/\"\" $%\"3G-%%VIEWG6$;F_bm$\"#6Fbbm;F^cm$FbemF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "The volume of revolution for region ABD rotated a round the " }{TEXT 355 1 "y" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[1] = Pi*Int((ln*y)^2,y = 0 .. ex p(1));" "6#/&%\"VG6#\"\"\"*&%#PiGF'-%$IntG6$*$*&%#lnGF'%\"yGF'\"\"#/F0 ;\"\"!-%$expG6#F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 114 "Q uoting the value of this integral which has been evaluated in the firs t worksheet on volumes of revolution (with " }{TEXT 356 1 "x" }{TEXT -1 13 " in place of " }{TEXT 357 1 "y" }{TEXT -1 10 ") we have " } {XPPEDIT 18 0 "V[1]=Pi*(exp(1)-2)" "6#/&%\"VG6#\"\"\"*&%#PiGF',&-%$exp G6#F'F'\"\"#!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "H ence the required volume of revolution obtained by rotating ODBC aroun d the " }{TEXT 358 1 "y" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "V[2]-V[1]=Pi*exp(1)-Pi*(exp(1)-2)" "6 #/,&&%\"VG6#\"\"#\"\"\"&F&6#F)!\"\",&*&%#PiGF)-%$expG6#F)F)F)*&F/F),&- F16#F)F)F(F,F)F," }{XPPEDIT 18 0 "``=2*Pi" "6#/%!G*&\"\"#\"\"\"%#PiGF' " }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 125 "Find the volume of the solid of revolution formed when the region in the first quadrant bounded by the section of the curve " } {XPPEDIT 18 0 "y = cos*x;" "6#/%\"yG*&%$cosG\"\"\"%\"xGF'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=Pi /2" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 24 " is rotated around \+ the " }{TEXT 304 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution" }{TEXT -1 4 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 786 "f := x -> cos(x):\np1 := plot(f(x),x=0..1.6,y=0..1.1,color=red,\n thickness=2,linestyle=1):\np2 := plot(f(-x),x=-1.6..0,y=0..1.1,colo r=red,\n thickness=1,linestyle=2):\na := 0: b := Pi/2: c := .9:\np3 \+ := plot(f(x),x=a..b,color=COLOR(RGB,.9,.9,.9),filled=true):\nh := eval f(Pi/20): r := .12:\np4 := plot([[seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i =0..40)],\n [seq([b*cos(i*h),r*b*sin(i*h)],i=0..40)],\n [[-b, 0],[-b,f(b)]]],linestyle=2,color=black):\np5 := plot([[b,0],[b,f(b)]], color=black):\np6 := plot([[seq([c*cos(i*h),f(c)+r*c*sin(i*h)],i=0..40 )],\n [seq([c*cos(i*h),r*c*sin(i*h)],i=0..40)],\n \+ [[c,0],[c,f(c)]],[[-c,0],[-c,f(c)]]],\n linestyle=1,color=COLOR( RGB,.7,.7,.7),thickness=2):\nplots[display]([p1,p2,p3,p4,p5,p6],tickma rks=[3,2],view=[-1.7..1.7,-.2..1.1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 516 209 209 {PLOTDATA 2 "60-%'CURVESG6&7S7$$\"\"!F)$\"\"\"F)7$$\"3Qmmm mYa([$!#>$\"3Y_9[K\">R***!#=7$$\"3%QLLLjQ?_'F/$\"3??56W!R(y**F27$$\"3B nmmmWkM**F/$\"3wG%\\V*>p]**F27$$\"3kmmmEr)pL\"F2$\"3A#)Q^Gjv5**F27$$\" 3GLLL.r()y;F2$\"3'yUs/I*Rf)*F27$$\"3mmmmwG&e*>F2$\"3\"=&f\"=%))[,)*F27 $$\"3')******HH1CBF2$\"3HA7Lf*\\6t*F27$$\"3_mmmO#)\\jEF2$\"3;rL\\/4QZ' *F27$$\"3&*******p\\%=+$F2$\"3[$H\\F]>Gb*F27$$\"3wLLLth()\\LF2$\"3!H!G *f!Q9W%*F27$$\"3#ommmCAkl$F2$\"3w^#z*oK%*Q$*F27$$\"3O+++?0_,SF2$\"3'f] z^r<+@*F27$$\"3o*******zN![VF2$\"3Wv%fB)z_p!*F27$$\"3'*******zu'>o%F2$ \"3YIy)G1NQ#*)F27$$\"3gmmmO%4_)\\F2$\"3!)o33\"*p!Hy)F27$$\"3!HLLLX$zX` F2$\"3](3&o,-$[g)F27$$\"3qLLL8aD^cF2$\"35tZ_pP@X%)F27$$\"3#)*******G!e 1gF2$\"3k)=$G^)Q'\\#)F27$$\"3NKLL8I5@jF2$\"3?T(R/PCy1)F27$$\"37+++!H%= mmF2$\"3aQ/z&)3IF(pF27$$\"3CmmmO9'[M)F2$\"3ai$)QR^e:nF27$$\"3%)******p!R>l)F 2$\"3G7\"Qz@D\\['F27$$\"3smmmE8f$)*)F2$\"3RY)eSmW*GiF27$$\"3A******f0A E$*F2$\"3'y>Q@P+t&fF27$$\"3o******>kTh'*F2$\"399nt)>!y%o&F27$$\"3S**** **\\ct&)**F2$\"3#H4VWf?]T&F27$$\"3&******fo$eM5!#<$\"3.KYZ6'Q)3^F27$$ \"39LLL\"QSp1\"Fau$\"3!oDZfwe!G[F27$$\"35+++g!)[,6Fau$\"3,<#oo\\%pAXF2 7$$\"3gmmm\"R$zK6Fau$\"3U3jA\"p<8C%F27$$\"3-+++)Q=q;\"Fau$\"3e!oo/n_*G RF27$$\"3;LLLU9A*>\"Fau$\"3#GPuv!H$3j$F27$$\"33+++8H)GB\"Fau$\"3c:LF27$$\"3KLLL`Jzl7Fau$\"3$o!R/A?'H+$F27$$\"3%)*****\\7Z-I\"Fau$\"39 4EU6qgsEF27$$\"3zmmm%RIML\"Fau$\"37(QkgMK9N#F27$$\"3immm!3ltO\"Fau$\"3 X')ROV+J??F27$$\"3WLLLq(=5S\"Fau$\"3Ol:?z5j*o\"F27$$\"3'******f,V>V\"F au$\"3;a:dJb2%Q\"F27$$\"3ULLL\"p&Qn9Fau$\"3R&pd!fUEK5F27$$\"3cmmmUg3* \\\"Fau$\"3a#Qsa)R)[;(F/7$$\"38+++H_)G`\"Fau$\"3#>o[^r,-z$F/7$$\"3++++ j`Bl:Fau$\"3[6&QM8N4c&!#?7$$\"33+++++++;Fau$!3a\"))G,B_*>HF/-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#F+-F$6&7S 7$$!33+++++++;FauFhz7$$!3MLLL`X7l:Fau$\"3))f`Q_IurcFdz7$$!3rmmm8'zZ`\" Fau$\"3C1M\")zE*3g$F/7$$!3LLLLbNl+:Fau$\"3!=B\\Q%o_3qF/7$$!3WLLL(G,jY \"Fau$\"3\"oJ\"))f(\\I/\"F27$$!3qmmm*G7@V\"Fau$\"3c&Qoq1*R#Q\"F27$$!3[ LLL7ZT+9Fau$\"39w.R,Xe&p\"F27$$!3;+++2Pfn8Fau$\"3w)\\'*>(32=?F27$$!3\\ LLLw,lL8Fau$\"3W0(*z(4'H\\BF27$$!3/+++.b\")*H\"Fau$\"33$)RlwfwwEF27$$! 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Using cylindrical shells the required volume i s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y] = 2*Pi*In t(x*f(x),x = 0 .. Pi/2);" "6#/&%\"VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG 6$*&%\"xGF*-%\"fG6#F0F*/F0;\"\"!*&F+F*F)!\"\"F*" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*cos*x ,x = 0 .. Pi/2);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*(%\"xGF'%$cosGF 'F-F'/F-;\"\"!*&F(F'F&!\"\"F'" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = sin*x],[du/dx = 1, dv/dx = cos*x]);" "6#-%*PIEC EWISEG6$7$/%\"uG%\"xG/%\"vG*&%$sinG\"\"\"F)F.7$/*&%#duGF.%#dxG!\"\"F./ *&%#dvGF.F3F4*&%$cosGF.F)F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*Pi" " 6#/%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 307 1 "[" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x;" "6#*(%\"xG\"\"\"%$sinGF%F$F%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``]);" "6#-%*P IECEWISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 2 " " } {XPPEDIT 18 0 "-Int(sin*x,x = 0 .. Pi/2);" "6#,$-%$IntG6$*&%$sinG\"\" \"%\"xGF)/F*;\"\"!*&%#PiGF)\"\"#!\"\"F1" }{TEXT -1 1 " " }{TEXT 317 1 "]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "(using the integrat ion by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-Int(v*` `(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"x G,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*Pi;" "6# /%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 324 1 "[" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x" "6#*(%\"xG\"\"\"%$sinGF%F$F%" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``]);" "6#-%*PIECEW ISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "``+cos*x;" "6#,&%!G\"\"\"*&%$cosGF%%\"xGF%F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``]);" "6#-%*PIECEWI SEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT 325 1 "]" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi* (Pi/2-1);" "6#/%!G*(\"\"#\"\"\"%#PiGF',&*&F(F'F&!\"\"F'F'F+F'" } {XPPEDIT 18 0 "`` = Pi*(Pi-2);" "6#/%!G*&%#PiG\"\"\",&F&F'\"\"#!\"\"F' " }{TEXT -1 1 " " }{TEXT 308 1 "~" }{TEXT -1 13 " 3.586419094." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> cos(x);\n2*Pi*Int(x*f(x),x=0..Pi/2);\nvalue(%);\nevalf(e valf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$cosG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&-%$c osG6#F,F&/F,;\"\"!,$*&F%!\"\"F'F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&,&*&F%!\"\"F'F&F&F&F*F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%4>ke$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 319 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 110 "Find the volume of the solid of revolution formed when the region in the first quadrant boun ded by the curve " }{XPPEDIT 18 0 "y = 1/(1+x^2);" "6#/%\"yG*&\"\"\"F &,&F&F&*$%\"xG\"\"#F&!\"\"" }{TEXT -1 14 " and the line " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 12 " is rotated " }}{PARA 0 "" 0 "" {TEXT -1 15 "(a) around the " }{TEXT 321 1 "x" }{TEXT -1 23 " axi s, (b) around the " }{TEXT 318 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "f := x -> 1/(1+x^ 2):\np1 := plot([f(x),-f(x)],x=0..1.4,y=-1.1..1.1,color=red,\n thick ness=[2,1],linestyle=[1,2]):\na := 0: b := 1:\np2 := plot(f(x),x=a..b, color=COLOR(RGB,.85,.85,.85),filled=true):\np3 := plot([[[a,0],[a,f(a) ]],[[b,0],[b,f(b)]]],color=black):\nh := evalf(Pi/20): r := .15:\np4 : = plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)],\n [seq ([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n linestyle=2,col or=black):\nplots[display]([p1,p2,p3,p4],tickmarks=[2,4],view=[-.2..1. 4,-1.1..1.1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 291 365 365 {PLOTDATA 2 " 6,-%'CURVESG6&7S7$$\"\"!F)$\"\"\"F)7$$\"3+LLL$e,;0$!#>$\"3G^+b\"R'p!** *!#=7$$\"3fmm;/Qy1dF/$\"3S6guP$Qv'**F27$$\"3!GLL$3R\"Gp)F/$\"3_P!G5r,] #**F27$$\"3KLL$etj)p6F2$\"3UH1r5'*)\\')*F27$$\"3cmmTlu,p9F2$\"3cu'R(4u v)y*F27$$\"37LL3n7PYy! 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" }}{PARA 0 "" 0 " " {TEXT -1 45 "Using vertical discs the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int(f(x)^2,x = 0 \+ .. 1);" "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$IntG6$*$-%\"fG6#F'\"\"#/F';\" \"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(1/((1+x^2)^2),x = 0 .. 1);" "6#/%!G*&%#PiG\"\"\"-%$I ntG6$*&F'F'*$,&F'F'*$%\"xG\"\"#F'F0!\"\"/F/;\"\"!F'F'" }{TEXT -1 12 " \+ ... " }{XPPEDIT 18 0 "PIECEWISE([x = tan*theta, `x =`*0*` impl ies `*theta*`=`*0],[dx = sec^2*theta*`.`*d*theta, `x =`*1*` implies ` *theta*`=`*``(Pi/4)]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%$tanG\"\"\"%&thet aGF+*.%$x~=GF+\"\"!F+%,~~implies~~GF+F,F+%\"=GF+F/F+7$/%#dxG*,%$secG\" \"#F,F+%\".GF+%\"dGF+F,F+*.F.F+F+F+%*~implies~GF+F,F+F1F+-%!G6#*&%#PiG F+\"\"%!\"\"F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Pi*Int(``(1/(sec^2*theta))^2*`.`*sec^2*theta,theta = 0 .. Pi/4);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$**-F$6#*&F'F'*&%$secG\"\" #%&thetaGF'!\"\"F1%\".GF'F0F1F2F'/F2;\"\"!*&F&F'\"\"%F3F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(s ec^2*theta/(sec^4*theta),theta = 0 .. Pi/4);" "6#/%!G*&%#PiG\"\"\"-%$I ntG6$*(%$secG\"\"#%&thetaGF'*&F,\"\"%F.F'!\"\"/F.;\"\"!*&F&F'F0F1F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(cos^2*theta,theta = 0 .. Pi /4)" "6#/%!G*&%#PiG\"\"\"-%$IntG6$*&%$cosG\"\"#%&thetaGF'/F.;\"\"!*&F& F'\"\"%!\"\"F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Pi*Int(``(1/2+cos*2*theta/2),theta = 0 .. Pi/4);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,&*&F'F'\"\"#!\"\"F'**%$cosGF'F/F'%& thetaGF'F/F0F'/F3;\"\"!*&F&F'\"\"%F0F'" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi;" "6#/%!G%#PiG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(theta/2+sin*2*theta/4);" "6#-%!G6#,&*&%&th etaG\"\"\"\"\"#!\"\"F)**%$sinGF)F*F)F(F)\"\"%F+F)" }{TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([Pi/4, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$*&% #PiG\"\"\"\"\"%!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(Pi/8+1/4);" "6#/%!G*&%#PiG\"\" \",&*&F&F'\"\")!\"\"F'*&F'F'\"\"%F+F'F'" }{TEXT -1 2 " " }{TEXT 322 1 "~" }{TEXT -1 15 " 2.019098714. 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= 2*Pi*Int(x/(1+x^2),x = 0 .. 1);" "6#/%!G*(\"\"#\" \"\"%#PiGF'-%$IntG6$*&%\"xGF',&F'F'*$F-F&F'!\"\"/F-;\"\"!F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*In t(2*x/(1+x^2),x = 0 .. 1);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$*(\"\"#F'%\"x GF',&F'F'*$F-F,F'!\"\"/F-;\"\"!F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*ln(1+x^2)" "6#/%!G*&%#PiG\"\" \"-%#lnG6#,&F'F'*$%\"xG\"\"#F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC EWISE([1, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*ln*2;" "6#/%!G*(%#PiG\"\"\"%#ln GF'\"\"#F'" }{TEXT -1 1 " " }{TEXT 323 1 "~" }{TEXT -1 13 " 2.17758609 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> 1/(1+x^2);\n2*Pi*Int(x*f(x),x=0..1);\nvalue (%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 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-1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "Pi*Int(25/(25-x^2),x = 0 .. 3) = 5* Pi*ln(2)" "6#/*&%#PiG\"\"\"-%$IntG6$*&\"#DF&,&F+F&*$%\"xG\"\"#!\"\"F0/ F.;\"\"!\"\"$F&*(\"\"&F&F%F&-%#lnG6#F/F&" }{TEXT -1 1 " " }{TEXT 345 1 "~" }{TEXT -1 18 " 10.88793045 (b) " }{XPPEDIT 18 0 "2*Pi*Int(5*x/s qrt(25-x^2),x = 0 .. 3) = 10*Pi;" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*( \"\"&F&%\"xGF&-%%sqrtG6#,&\"#DF&*$F-F%!\"\"F4/F-;\"\"!\"\"$F&*&\"#5F&F 'F&" }{TEXT -1 1 " " }{TEXT 346 1 "~" }{TEXT -1 13 " 31.41592654 " }}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }} {PARA 0 "" 0 "" {TEXT -1 110 "Find the volume of the solid of revoluti on formed when the region in the first quadrant bounded by the curve \+ " }{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 14 " and the line " }{XPPEDIT 18 0 "x = Pi/2;" "6#/%\"xG*&%#PiG\"\"\"\"\"# !\"\"" }{TEXT -1 23 " is rotated around the " }{TEXT 347 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " 2*Pi*Int(x*sin(x),x = 0 .. Pi/2)=2*Pi" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG 6$*&%\"xGF&-%$sinG6#F,F&/F,;\"\"!*&F'F&F%!\"\"F&*&F%F&F'F&" }{TEXT -1 1 " " }{TEXT 348 1 "~" }{TEXT -1 13 " 6.283185307 " }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Find the volume of the solid of rev olution formed when the region in the first quadrant bounded by the cu rve " }{XPPEDIT 18 0 "y = ln(x);" "6#/%\"yG-%#lnG6#%\"xG" }{TEXT -1 6 ", the " }{TEXT 339 1 "x" }{TEXT -1 19 " axis and the line " } {XPPEDIT 18 0 "x = exp(1);" "6#/%\"xG-%$expG6#\"\"\"" }{TEXT -1 23 " i s rotated around the " }{TEXT 338 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*Pi*Int(x*ln(x),x = 1 .. exp(1)) = 1/2;" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&-%#lnG6# F,F&/F,;F&-%$expG6#F&F&*&F&F&F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Pi*(exp(2)+1);" "6#*&%#PiG\"\"\",&-%$expG6#\"\"#F%F%F%F%" }{TEXT -1 2 " " }{TEXT 340 1 "~" }{TEXT -1 18 " 13.17749851 or " }{XPPEDIT 18 0 "Pi*exp(2)-Pi*Int(exp(2*y),y = 0 .. 1) = Pi*exp(2)-1/2;" "6#/,&*&%#P iG\"\"\"-%$expG6#\"\"#F'F'*&F&F'-%$IntG6$-F)6#*&F+F'%\"yGF'/F3;\"\"!F' F'!\"\",&*&F&F'-F)6#F+F'F'*&F'F'F+F7F7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*(exp(2)-1)-1/2;" "6#,&*&%#PiG\"\"\",&-%$expG6#\"\"#F&F&!\"\"F&F& *&F&F&F+F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*(exp(2)+1)" "6#*&%#Pi G\"\"\",&-%$expG6#\"\"#F%F%F%F%" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "vol of rev for region by shell method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2286 "f := x -> 2-sin(x)+sqrt(x) /7+x/7:\na := 1: b := 5:\nc := 2.9: d := 3.1: m := (c+d)/2: \nq := eva lf(Pi/40): r := .25:\ndarkblue := COLOR(RGB,.5,0,.7):\np1 := plottools [arrow]([0,-.1],[0,4.5],0,.1,.02,arrow,color=black):\np1a := plottools [arrow]([-5.5,0],[5.5,0],0,.1,.01,arrow,color=black):\np2 := plot([[[a ,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot([[[-a,0],[-a ,f(a)]],[[-b,0],[-b,f(b)]]],linestyle=2,color=black):\np4 := plot([[[m ,0],[m,-.3]],[[c,f(m)],[c,f(m)+1]],\n [[d,f(m)],[d,f(m)+1]],[ [d,f(m)],[d+.7,f(m)]],\n [[d,0],[d+.7,0]]],\n color =darkblue,linestyle=2):\np5 := plot(f(x),x=0.7..5.5,color=red,thicknes s=2):\np6 := plot(f(-x),x=-5.5..-0.7,color=red,\n line style=2,thickness=1):\n\np7 := plot([[seq([c*cos(i*q),f(m)+r*c*sin(i*q )],i=0..80)],\n [seq([d*cos(i*q),f(m)+r*d*sin(i*q)],i=0..80)],\n \+ [seq([d*cos(i*q),r*d*sin(i*q)],i=40..80)],\n [[-d,0],[-d,f(m )]],[[d,0],[d,f(m)]]],color=darkblue):\n\np8 := plot([[seq([c*cos(i*q) ,r*c*sin(i*q)],i=0..80)],\n [seq([d*cos(i*q),r*d*sin(i*q)],i=0.. 40)],\n [[-c,0],[-c,f(m)]],[[c,0],[c,f(m)]]],\n linestyle= 2,color=darkblue):\n\np9 := plot([[seq([a*cos(i*q),f(a)+r*a*sin(i*q)], i=0..80)],\n [seq([b*cos(i*q),f(b)+r*b*sin(i*q)],i=0..80)],\n \+ [seq([a*cos(i*q),r*a*sin(i*q)],i=0..80)],\n [seq([b*cos(i*q), r*b*sin(i*q)],i=0..80)]],\n linestyle=2,color=black):\n\np10 := \+ plot(f(x),x=a..b,filled=true,color=COLOR(RGB,.95,.95,.95)):\np11 := pl ot(f(-x),x=-b..-a,filled=true,color=COLOR(RGB,.98,.98,.98)):\np12 := p lottools[arrow]([d+.4,f(m)/2+.3],[d+.4,f(m)],0,.05,.07,arrow,color=dar kblue):\np13 := plottools[arrow]([d+.4,f(m)/2-.3],[d+.4,0],0,.05,.07,a rrow,color=darkblue):\np14 := plottools[arrow]([d+.15,f(m)+.8],[d,f(m) +.8],0,.1,.3,arrow,color=darkblue):\np15 := plottools[arrow]([c-.15,f( m)+.8],[c,f(m)+.8],0,.1,.3,arrow,color=darkblue):\nt1 := plots[textplo t]([[a,-.2,'a'],[b,-.2,'b'],[-.1,4.3,`y`],\n [5.4,-.2,`x`]] ,color=black):\nt2 := plots[textplot]([4.1,4.2,`y = f(x)`],color=red): \nt3 := plots[textplot]([[m,-.4,`x`],[m+.05,3.7,`x`],\n [d+.4,f(m)/ 2,`f(x)`]],color=darkblue):\nt4 := plots[textplot]([[m-.05,3.7,'d']],f ont=[SYMBOL,10],color=darkblue):\nplots[display]([p1,p1a,p2,p3,p4,p5,p 6,p7,p8,p9,p10,p11,\n p12,p13,p14,p15,t1,t2,t3,t4],axes=none);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "pictures of cylindric al shell " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 2930 "r := 1.1: h := 2: rp := 1:\ndarkblue := COLOR(RGB ,.5,0,.7):\np1 := plot([r*cos(t),r*0.3*sin(t),t=-Pi..0],color=darkblue ):\np2 := plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],color=darkblue):\nA 1:=[r,0]: B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\nAp:=[rp,0]: Bp:=[rp,h]: \+ Cp:=[-rp,h]: Dp:=[-rp,0]:\np3 := plot([[A1,B1],[D1,C1]],color=darkblue ):\na := evalf(Pi/60):\nc1 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-12..0 ):\nc2 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27..-12):\nc3 := seq([r*c os(j*a),r*0.3*sin(j*a)],j=-33..-27):\nc4 := seq([r*cos(j*a),r*0.3*sin( j*a)],j=-48..-33):\nc5 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-60..-48): \nc6 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..12):\nc7 := seq([r*cos( j*a),h-r*0.3*sin(j*a)],j=12..27):\nc8 := seq([r*cos(j*a),h-r*0.3*sin(j *a)],j=27..33):\nc9 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=33..48):\nc 10 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=48..60):\np4 := plots[polygo nplot]([c1,c6],\n style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\np 5 := plots[polygonplot]([c2,c7],\n style=patchnogrid,color=COLOR(RG B,.95,.91,1)):\np6 := plots[polygonplot]([c3,c8],\n style=patchnogr id,color=COLOR(RGB,.953,.923,1)):\np7 := plots[polygonplot]([c4,c9],\n style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := plots[polygo nplot]([c5,c10],\n style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\n pp := seq([[r*cos(2*j*a),r*0.3*sin(2*j*a)+h],\n [r*cos(2*(j+ 1)*a),r*0.3*sin(2*(j+1)*a)+h],\n [rp*cos(2*(j+1)*a),rp*0.3*s in(2*(j+1)*a)+h],\n [rp*cos(2*j*a),rp*0.3*sin(2*j*a)+h]],j=0 ..60):\np9 := plots[polygonplot]([pp],\n style=patchnogrid,color=CO LOR(RGB,.953,.933,1)):\np10 := plot([[A1,[r+.25,0]],[B1,[r+.25,h]]],\n color=darkblue,linestyle=2):\np11 := plottools[arro w]([r+.19,h/2-.13],[r+.19,0],\n 0,.04,.1,arrow,color=darkblue ):\np12 := plottools[arrow]([r+.19,h/2+.13],[r+.19,h],\n 0,.0 4,.1,arrow,color=darkblue):\np13 := plottools[arrow]([rp/2+.1,h],[(rp+ r)/2,h],\n 0,.07,.08,arrow,color=darkblue):\np14 := plot([[0, h],[rp/2-0.1,h]],color=darkblue):\np15 := plot([[0,h]],style=point,sym bol=cross,color=darkblue):\np16 := plot([rp*cos(t),rp*0.3*sin(t),t=0.. 2*Pi],\n linestyle=2,color=darkblu e):\np17 := plot([rp*cos(t),rp*0.3*sin(t)+h,t=0..2*Pi],\n \+ linestyle=1,color=darkblue):\np18 := plot([[Ap ,[rp,h+0.4]],[Dp,Cp],[B1,[r,h+0.4]],\n [[(rp+r)/2,h-0.2],[(rp+r)/2, h+0.2]]],linestyle=2,color=darkblue):\np19 := plottools[arrow]([rp-.1, h+.3],[rp,h+.3],\n 0,.06,.3,arrow,color=darkblue):\np20 := pl ottools[arrow]([r+.1,h+.3],[r,h+.3],\n 0,.06,.3,arrow,color=d arkblue):\nt1 := plots[textplot]([[r+.2,h/2,`f(x)`],[rp+.07,h+.5,'`x`' ],\n [rp/2,h,'`x`']],font=[HELVETICA,10],color=darkblue):\n t2 := plots[textplot]([[rp+.03,h+.5,'d']],\n font =[SYMBOL,10],color=darkblue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8 ,p9,\n p10,p11,p12,p13,p14,p15,p16,p17,p18,p19,p20,t1,t2],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1524 "c := 7.2: h := 2:\ndarkblue := COLOR(RGB,.5,0,.7): \nA2 := [0,0]: B2 := [0,h]: C2 := [c,h]: D2 := [c,0]:\nB3 := [.07,h+.0 7]: C3 := [c+.07,h+.07]: D3 := [c+.07,.07]: \np1 := plot([[A2,B2,C2,D2 ,A2],[B2,B3,C3,C2],[C3,D3,D2]],color=darkblue):\np2 := plots[polygonpl ot]([A2,B2,C2,D2],\n style=patchnogrid,color=COLOR(RGB,.95,.91 ,1)):\np3 := plots[polygonplot]([B2,B3,C3,C2],\n style=patchno grid,color=COLOR(RGB,.953,.923,1)):\np4 := plots[polygonplot]([C2,C3,D 3,D2],\n style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\np5 := plot([[A2,[0,-.3]],[D2,[c,-.3]],[A2,[-.5,0]],\n [B2,[-.5,h]],[ C2,[c+.7,h]],[C3,[c+.9,h+.07]]],\n color=darkblue,linesty le=2):\np6 := plottools[arrow]([c/2-.3,-.2],[0,-.2],\n 0,.08, .03,arrow,color=darkblue):\np7 := plottools[arrow]([c/2+.3,-.2],[c,-.2 ],\n 0,.08,.03,arrow,color=darkblue):\np8 := plottools[arrow] ([-.3,h/2+.13],[-.3,h],\n 0,.1,.07,arrow,color=darkblue):\np9 := plottools[arrow]([-.3,h/2-.13],[-.3,0],\n 0,.1,.07,arrow, color=darkblue):\np10 := plottools[arrow]([c+.3,h-.2],[c+.5,h],\n \+ 0,.07,.3,arrow,color=darkblue):\np11 := plottools[arrow]([c+.77,h +.27],[c+.57,h+.07],\n 0,.07,.3,arrow,color=darkblue):\nt1 := plots[textplot]([[c/2,-.2,'`2 x`'],[c+.5,h+.2,`x`],\n [-.3, h/2,`f(x)`]],font=[HELVETICA,10],color=darkblue):\nt2 := plots[textplo t]([[c/2+.01,-.2,`p`],[c+.42,h+.2,'d']],\n \+ font=[SYMBOL,10],color=darkblue):\nplots[display]([p1,p2,p3,p4,p5,p6,p 7,p8,p9,p10,p11,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 16 "example picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "f := x -> exp(x):\np1 := pl ot(f(x),x=0..1.1,y=0..3,color=red,thickness=2):\na := 0: b := 1:\np2 : = plot(f(x),x=a..b,color=COLOR(RGB,.85,.85,.85),filled=true):\np3 := p lot([[b,0],[b,f(b)],[0,f(b)]],linestyle=[1,3],color=black):\nt1 := plo ts[textplot]([[-.08,2.7,`A`],[1.08,2.7,`B`],[1.08,-.13,`C`],\n [-.07, -.13,`O`],[.07,1.25,`D`]],color=black):\nplots[display]([p1,p2,p3,t1], xtickmarks=[1=`1`],\n ytickmarks=[1=`1`,2=`2`,3=`3`],view=[-.08..1.1 ,-.13..3]);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 22 "task question picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "f := x -> 5/sqrt(25-x^2): \np1 := plot(f(x),x=0..5.1,y=0..3,color=red,thickness=2):\na := 0: b : = 3:\np2 := plot(f(x),x=a..b,color=COLOR(RGB,.85,.85,.85),filled=true) :\np3 := plot([[[b,0],[b,f(b)]],[[5,0],[5,3]]],linestyle=[1,3],color=b lack):\nplots[display]([p1,p2,p3],tickmarks=[6,4]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }