{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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