{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Module 9 : Integral Calculus" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 26 "903 : Sol ids of Revolution" }}{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 256 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 202 "In this module, we will examine solids created b y rotating functions around the x and y axes, creating three dimensio nal colored graphs and computing these volumes symbolically, with nume rical results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 252 "In this project we will use the following command packages. Type and execute this lin e before begining the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to \+ any point in the worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "re start; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name \+ changecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 432 "Note that this project creates many larg e graphs. Depending on the capabilities of computer system you may exp erience memory problems. You can overcome these in two ways. Whenever \+ you plan to save your file, first remove the output. [tell them how to do this!!] When you re-enter your program, do not execute worksheet, \+ but rather just re-execute the commands restart; with(plots): and then go to where you want to continue working." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________ ______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 30 "A. Revolving Around The X Axis" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________________________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Lets see what it looks like to revolve functions around the x-axis. First lets define a func tion and see what its graph looks like." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " f := x -> 2 - sin(x); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%& arrowGF(,&\"\"#\"\"\"-%$sinG6#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " plot( f(x),x = 0..2*Pi, y = 0..3);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7en7$$\"\"!F)$ \"\"#F)7$$\"3i]cC&eb&p8!#=$\"3c[J.#=sM'=!#<7$$\"3YqR*pd)>hDF/$\"3-A\"o X5rmu\"F27$$\"3Zs[E^dK,RF/$\"3\"\\s#Go*)o>;F27$$\"3ab^NehL]_F/$\"3w4@- 2zv)\\\"F27$$\"35*QhW&\\$Hf'F/$\"3#3V2TVTuQ\"F27$$\"3zS11.fpPyF/$\"39o G4Jd/%H\"F27$$\"3upr'fwtl7*F/$\"325mB'4o)37F27$$\"3!QRa&4L&f/\"F2$\"3F b\"RsQ(fM6F27$$\"3YjXwg<#)y6F2$\"3mjH&)\\M%e2\"F27$$\"36qGW%H$\\:8F2$ \"3J1$fs=8C.\"F27$$\"3SH22vMov8F2$\"3!\\+&e.U(*=5F27$$\"3$*)e)pbO(eV\" F2$\"3_Sp.k#)345F27$$\"3/]+3VNj.:F2$\"3R=f1sXD-5F27$$\"396:YIMRr:F2$\" 3)*)Qm#y,++5F27$$\"3Z'z]kaJ%R;F2$\"3Hi[=sWN-5F27$$\"3!=3SCmpuq\"F2$\"3 L[Euq_K45F27$$\"33u?N%*p.tF2$\"3wdZ%[M8R2\"F27$$\"3#=uye<)G*4#F2$\"3H4! 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To create a three dimensional view of the resu lting solid of revolution, we will use three dimensional coordinates ( x,y,z). The z axis will measure how high off of the xy plane the surfa ce is. Any given point on the original graph (x,f(x)) will be revolved around the x-axis to create a circle of radius f(x). Each point on th e circle is of the form (a,b,c) where a = x, b2 + c2 = f(x)2. See the \+ diagram below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {METAFILE 316 227 227 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "This can be a little confusing so dont wo rry about it too much. \n\n" }}{PARA 0 "" 0 "" {TEXT -1 91 "Using this idea, we can revolve a function around the x-axis, by using this Mapl e command :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 135 " implicitplot3d( \{y^2 + z^2 = f(x)^2\}, x = 0..2* Pi, y = -3..3, z =-3..3, \n\011style = patchnogrid, orientation = [-50 ,70], axes = boxed );" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%+ISOSURFACEG6#X0%)anythingG6\"6\"_[[[[]\\q[ez\"+\"+\" +\"%0000000000000000C008000000000000C008000000000000402C00000000000000 00000000000000C008000000000000C002AAAAAAAAAAAB4024E38E38E38E3A00000000 00000000C008000000000000BFFAAAAAAAAAAAAC401F1C71C71C71C800000000000000 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02AAAAAAAAAAA94024E38E38E38E25401921FB54442D114007FFFFFFFFFFFE4007FFFF FFFFFFFE402BFFFFFFFFFFEB-%&STYLEG6#%,PATCHNOGRIDG-%*AXESSTYLEG6#%$BOXG -%+AXESLABELSG6%%\"xG%\"yGQ!6\"-%+PROJECTIONG6%$!#]\"\"!$\"#qF>\"\"\" " 1 3 0 1 10 0 2 1 1 2 2 1.000000 70.000000 -50.000000 1 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "Click anywhere on the graph and you will see a black border appear. While h olding the mouse down and dragging you will see the graph rotate in th ree dimensional space. Please inspect this graph from all angles. You \+ can look over, under, and through it!" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "Here is another example which you can use as a template for the exercise that follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " f:= x -> 1- x; a:= -1; b := 1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6# %\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " plot( f(x), \+ x =a..b);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%- %'CURVESG6$7S7$$!\"\"\"\"!$\"\"#F*7$$!3ommm;p0k&*!#=$\"3ymmm\"p0k&>!#< 7$$!3wKL$3F37$$!3mmmmT%p\"e()F0$\"3cmm;Wp\"e(=F37$ $!3:mmm\"4m(G$)F0$\"3hmm;4m(G$=F37$$!3\"QLL3i.9!zF0$\"3QLL3i.9!z\"F37$ $!3\"ommT!R=0vF0$\"3emmT!R=0v\"F37$$!3u****\\P8#\\4(F0$\"3)****\\P8#\\ 4s%HaF0$\"3mmm;>s% Ha\"F37$$!3Q+++]$*4)*\\F0$\"3/+++N*4)*\\\"F37$$!39+++]_&\\c%F0$\"3-+++ 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Revolvin g Around The Y Axis" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________ _________________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 280 "We can also revolve curves around the y axis. \+ However to accomplish this in Maple, we need to alter our scheme sligh tly to create a function x = g(y) which happens to be the inverse func tion of y = f(x). 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-20.000000 -90.000000 1 0 "Curve 1" }}}}{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 83 "_________ ______________________________________________________________________ ____" }}{PARA 4 "" 0 "" {TEXT -1 36 "C. Revolving A Region Between Cur ves" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________________ _________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "We can also revolve a region which is between two different fu nctions. This is slightly more complicated. 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013FEFFFFFFFFFFFFF3FE1C71C71C71C713FD3C0CA4587E6AC3FF00000000000013FEF FFFFFFFFFFFF3FE8E38E38E38E383FE35BA781948B0A3FF00000000000013FEFFFFFFF FFFFFF3FEFFFFFFFFFFFFF3FEFFFFFFFFFFFFA-F+6#%)ZSHADINGG-F/6#%,PATCHNOGR IDG-%+AXESLABELSG6&%\"xG%\"yGQ!6\"%(DEFAULTG" 1 2 0 1 10 0 2 1 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "In this display you \+ can see that the inner and outer functions are shaded differently and \+ by turning the graph you can get a visual idea of the thickness of the walls of the cup you see." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 83 "_______________________________________________________ ____________________________" }}{PARA 4 "" 0 "" {TEXT -1 18 "D. The Di sk Method" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________ _______________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 462 "In this section well compute the volume generated by rev olving a function around the x-axis using the disk method. In this met hod, we imagine that we cut the area to be rotated into thin vertical \+ slices of size y by delta x, rotate these slices to form a disk which \+ you might think of a thin wafer or coin, then sum these up using integ ration. The formula is V = y2 dx.\n\nHere is an example. We will rota te y = 1 + x*sin(x) between x = /4 and around the x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " f:= x -> 1 + x*sin(x); a := Pi/4; b := Pi;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&9$F --%$sinG6#F/F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,$%#Pi G#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG%#PiG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " plot( \{f(x),0\}, x = a..b) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6 $7S7$$\"3o***z'\\;)R&y!#=$\"\"!F,7$$\"3#)4q9%*\\cn$)F*F+7$$\"3gMD0;6V9 ))F*F+7$$\"3*f,/k!)ypJ*F*F+7$$\"3XkF14x&G#)*F*F+7$$\"3_\\-wDKjK5!#\"oWw7\"F:FrC,%*p;FFF\">fD#FIYHF$\"3 _V#*[%4d:&>F<7$FA$\"3F1@iu&>(=?F<7$FD$\"3?_GF<7$Fdp$ \"3T?>'3N>%=GF<7$Fgp$\"3Ig6l%R)*4\"GF<7$Fjp$\"3oP6(f!)Hiz#F<7$F]q$\"3] `$)HlliuFF<7$F`q$\"3SG'yYY^ou#F<7$Fcq$\"3F3H>9*fzq#F<7$Ffq$\"33!)o^6Vp lEF<7$Fiq$\"3,s(HJNZGh#F<7$F\\r$\"3'zj#[M&Q!eDF<7$F_r$\"3y4!=v,u0\\#F< 7$Fbr$\"3vKbH?=\"*>CF<7$Fer$\"3^y$\\bKW'QBF<7$Fhr$\"3mvX/k%Q>D#F<7$F[s $\"39**Gv0Je`@F<7$F^s$\"3)z6y!)pZ;0#F<7$Fas$\"3w\"f!y%*oCS>F<7$Fds$\"3 SRJ#GJ0G#=F<7$Fgs$\"3$G%f\\St&*3 " 0 "" {MPLTEXT 1 0 29 " Int( Pi*f(x)^2, x = a..b); \n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%#PiG\"\"\"),&F(F(*&%\"xGF( -%$sinG6#F,F(F(\"\"#F(/F,;,$F'#F(\"\"%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " value(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$ )%#PiG\"\"#\"\"\"#\"\"&F'*&#\"#@\"$G\"F()F&\"\"%F(F(*&#F(\"#kF()F&\"\" $F(F(*&-%%sqrtG6#F'F(F&F(!\"\"*(#F(F0F(F%F(F7F(F(*&#F(\"\")F(F&F(F:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<8NzR!\")" }}}{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 83 "_______________ ____________________________________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 20 "E. The Washer Method" }}{PARA 0 "" 0 " " {TEXT -1 83 "_______________________________________________________ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 275 "The disk metho d we saw in the last section works only if the slice being rotated con tinues without interruption from the x-axis to the curve, and wont wor k if there are gaps. Consider this example where we plan to rotate the area between these two functions around the x-axis" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Vertical slices inters ect the area to be rotated between these functions but do not continue down to the x-axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " f := x -> x^(1/8);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$#\"\"\" \"\")F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " g := x -> x^8;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operat orG%&arrowGF(*$)9$\"\")\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " a := 0; b := 1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"aG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " plot( \{f(x), g(x)\}, x = a ..b);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV ESG6$7W7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3o3)GjkNc4&!#J7$$\"3[LL$e9ui2% F-$\"3/kbHJ#pEi(!#H7$$\"3nmmm\"z_\"4iF-$\"309k)yw?$4A!#F7$$\"3[mmmT&ph N)F-$\"3?lo%pUdrP#!#E7$$\"3CLLe*=)H\\5!#=$\"3]sBdePdp9!#D7$$\"3gmm\"z/ 3uC\"FF$\"3/j\"Hp'RIieFI7$$\"3%)***\\7LRDX\"FF$\"3QYlL8Hj\")>!#C7$$\"3 ]mm\"zR'ok;FF$\"3!RF=%[XQ(*eFT7$$\"3w***\\i5`h(=FF$\"3O&\\q!HP8N:!#B7$ $\"3WLLL3En$4#FF$\"35yVXl?2#p$Fin7$$\"3qmm;/RE&G#FF$\"3@b:Z=WfQuFin7$$ \"3\")*****\\K]4]#FF$\"30$Q!RV`_I:!#A7$$\"3$******\\PAvr#FF$\"3'H>JQw, V(HFio7$$\"3)******\\nHi#HFF$\"3'3)4m'H*4w`Fio7$$\"3jmm\"z*ev:JFF$\"31 )))Fio7$$\"3?LLL347TLFF$\"3Ue>-:(yGb\"!#@7$$\"3,LLLLY.KNFF$\"31 XQE\\@9ACF^q7$$\"3w***\\7o7Tv$FF$\"3A(R$RyP5XRF^q7$$\"3'GLLLQ*o]RFF$\" 3Pr\">!Hc[MfF^q7$$\"3A++D\"=lj;%FF$\"3_rQpy;Vz!*F^q7$$\"31++vV&RFhr7$$\"3GLLeR\"3Gy%FF $\"3(efXOh!>QFFhr7$$\"3cmm;/T1&*\\FF$\"3#[\\y#Hq]vQFhr7$$\"3&em;zRQb@& FF$\"3AK#))[89^Z&Fhr7$$\"3\\***\\(=>Y2aFF$\"3NXQKx8^5tFhr7$$\"39mm;zXu 9cFF$\"3S&\\%pW2Mx)*Fhr7$$\"3l******\\y))GeFF$\"3h(y&>(QcDL\"F-7$$\"3' *)***\\i_QQgFF$\"3qsN)3+Hvw\"F-7$$\"3@***\\7y%3TiFF$\"3/7$))[Gp=I#F-7$ $\"35****\\P![hY'FF$\"3gvY]0I3cIF-7$$\"3kKLL$Qx$omFF$\"3ri5ko8')4RF-7$ $\"3!)*****\\P+V)oFF$\"3qFR4(=,_/&F-7$$\"3?mm\"zpe*zqFF$\"3UrP]Aa98jF- 7$$\"3%)*****\\#\\'QH(FF$\"3tLS\"p%)Q0,)F-7$$\"3GKLe9S8&\\(FF$\"3[7-@& *oWf**F-7$$\"3R***\\i?=bq(FF$\"3BZ[R7)QGC\"FF7$$\"3\"HLL$3s?6zFF$\"3mz 5j>BTM:FF7$$\"3a***\\7`Wl7)FF$\"3whgnj8;->FF7$$\"3#pmmm'*RRL)FF$\"3/?h eS^.FBFF7$$\"3Qmm;a<.Y&)FF$\"3-S!\\\"y5AXGFF7$$\"3=LLe9tOc()FF$\"3_?O: EN9cMFF7$$\"3u******\\Qk\\*)FF$\"3+i6p=>s:TFF7$$\"3CLL$3dg6<*FF$\"3\"[ ()*\\p\"))[+&FF7$$\"3y***\\(oTAq#*FF$\"3#*z'pY%[3aaFF7$$\"3ImmmmxGp$*F F$\"3E2Ayk*p\"QfFF7$$\"3sK$eRA5\\Z*FF$\"3M6MFZTK&\\'FF7$$\"3A++D\"oK0e *FF$\"3q5\"z+q$p(4(FF7$$\"3C+++]oi\"o*FF$\"3>v0uEkX>xFF7$$\"3A++v=5s#y *FF$\"3i0\"zacl$)Q)FF7$$\"35+]P40O\"*)*FF$\"3C'*RZ1GAj\"*FF7$$\"\"\"F) F`\\l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7gnF'7$$\"3/LL3x1h6oF^q$\"3l Cn9;PN>SFF7$$\"3immTN@Ki8Fhr$\"3_U[x2j8$Q%FF7$$\"3#***\\7.K[V?Fhr$\"3e SI*pR85h%FF7$$\"3ALL$3FWYs#Fhr$\"3;7V&p.W)zZFF7$$\"3%)***\\iSmp3%Fhr$ \"3v2S%[wX$G]FF7$$\"3WmmmT&)G\\aFhr$\"3/*fKK)oX7_FF7$$\"3m****\\7G$R<) Fhr$\"3%3*p2R*\\M[&FF7$$\"3ILLL3x&)*3\"F-$\"39(R\\t`CUo&FF7$$\"3$***** \\ilyM;F-$\"3'RHY-cW(zfFF7$F+$\"3u9^=73p)>'FF7$$\"3.++D\"y%*z7$F-$\"3] F*Qz>u\\['FF7$F2$\"3P:zi3:@.nFF7$F8$\"3/9aAfXFlqFF7$F>$\"3%*[1))3.[KtF F7$FD$\"3!QsU1i&=WvFF7$FK$\"3^8rCVO04xFF7$FP$\"3%G$pCmq;dyFF7$FV$\"3Ay ^![(Q?#*zFF7$Fen$\"3*)o(*>Y9d7\")FF7$F[o$\"3pIb!G[xXA)FF7$F`o$\"3&[W\\ ut\"4:$)FF7$Feo$\"30gea[OO4%)FF7$F[p$\"3U5tnPy6(\\)FF7$F`p$\"3aV>lAY2w &)FF7$Fep$\"3N%=jKF:Ok)FF7$Fjp$\"3^;vG\\%)R>()FF7$F`q$\"3CamRJR\"*FF7$Fds$\"3Y/ f#RQ3*o\"*FF7$Fis$\"3 " 0 "" {MPLTEXT 1 0 38 " Int( Pi*(f(x)^2-g (x)^2), x = a..b); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%# PiG\"\"\",&*$)%\"xG#F(\"\"%F(F(*$)F,\"#;F(!\"\"F(/F,;\"\"!F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " value(%); \n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$%#PiG#\"#j\"#&)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bXZ GB!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________________ _______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 31 "F. The Cylindrical Shell Method" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________________________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "In this section well c ompute the volume created by rotating a function around the y axis usi ng the cylindrical shell method. The formula is V = 2xy dx. Here is a n example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 " f:= x -> sqrt(abs(x^2 - 2 )); a := 0;b:= 4;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG F(-%%sqrtG6#-%$absG6#,&*$)9$\"\"#\"\"\"F7F6!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " plot( \{f(x), 0\}, x = a..b);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>F(7$$\"3RLLLe'40j\"!# =F(7$$\"3mmmm;6m$[#F1F(7$$\"3fmmm;yYULF1F(7$$\"3%HLL$eF>(>%F1F(7$$\"3Q mmm\">K'*)\\F1F(7$$\"3P*****\\Kd,\"eF1F(7$$\"3-mmm\"fX(emF1F(7$$\"3.** ***\\U7Y](F1F(7$$\"3'QLLLV!pu$)F1F(7$$\"3xmmm;c0T\"*F1F(7$$\"3#******* H,Q+5!#FPF(7$$\"3immmTc-)*>FPF(7$$\"3Mmm;f`@'3# FPF(7$$\"3y****\\nZ)H;#FPF(7$$\"3YmmmJy*eC#FPF(7$$\"3')******R^bJBFPF( 7$$\"3f*****\\5a`T#FPF(7$$\"3o****\\7RV'\\#FPF(7$$\"3k*****\\@fke#FPF( 7$$\"3/LLL`4NnEFPF(7$$\"3#*******\\,s`FFPF(7$$\"3[mm;zM)>$GFPF(7$$\"3$ *******pfa:Si%\\]8FP7$F<$\"3q$frw0nKK\"FP7$F?$\"3+EJW^*[$*G\"FP7$FB$ \"3o]F!\\?UwC\"FP7$FE$\"3')e@BI#p')>\"FP7$FH$\"3h*>osK\"eR6FP7$FK$\"3Q +P-%yy!z5FP7$FN$\"3iMAYas>'***F17$FR$\"3W01#y45m/*F17$FU$\"37QaA&GJp$z F17$FX$\"3q_fKl&*y$o'F17$$\"33++Dh`P\"H\"FP$\"3eBpnyw(\\w&F17$Fen$\"3% >d#)R$[*\\i%F17$$\"3MLLL36ju8FP$\"3fdY+m()[ALF17$Fhn$\"3m'zzHc\"[!H'F- 7$$\"3mmm\"HYHsX\"FP$\"3H8F]3'3X^$F17$F[o$\"3U6&RayP\"\\]F17$$\"3_mm\" HTg4a\"FP$\"3?#Q9-]A,7'F17$F^o$\"3w*[IBqZ<0(F17$Fao$\"38J#y#4=*o\"))F1 7$Fdo$\"3mpH*\\ng&G5FP7$Fgo$\"3-@W5i&e$o6FP7$Fjo$\"37$HJ`iD%)G\"FP7$F] p$\"3N&zIr,?9T\"FP7$F`p$\"3_;P`\"FP7$Fcp$\"3!y)*oKI8mj\"FP7$Ffp$ \"3%zj)>MEsWFP7$F_q$\"3=\\% 3?%oAd?FP7$Fbq$\"3IOZZtzel@FP7$Feq$\"3O-20hPehAFP7$Fhq$\"3UXSE;>$GO#FP 7$F[r$\"3C#GB!3af`CFP7$F^r$\"3)oIX)fm(=b#FP7$Far$\"31,.!G5WNk#FP7$Fdr$ \"3L]$)z.JhQFFP7$Fgr$\"3f#)yYCC*3$GFP7$Fjr$\"3(*e?9X6'o#HFP7$F]s$\"3TR 2Nw'H(=IFP7$F`s$\"3'p3$)*))3;7JFP7$Fcs$\"3VMf8iiM/KFP7$Ffs$\"3AZE/1Xn) G$FP7$Fis$\"3uXFx!37\\Q$FP7$F\\t$\"3U.@M>QkqMFP7$F_t$\"3!p;yt2@ " 0 "" {MPLTEXT 1 0 31 " Int( 2*Pi*x*f(x), x = a..b); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$,$*(%#PiG\"\"\"%\"xGF)-%%sqrtG6#-%$absG6#,&*$)F*\"\"#F)F)F4 !\"\"F)F4/F*;\"\"!\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " value(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sqrtG6#\"#9\" \"\"%#PiGF)#\"#G\"\"$*(#\"\"%F-F)-F&6#\"\"#F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'o\\j:\"!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }