{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "Courier" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 8 "MATH 149" }}{PARA 256 "" 0 "" {TEXT 288 0 "" }{TEXT 258 0 "" } {TEXT 289 0 "" }{TEXT 259 23 "LABORATORY ASSIGNMENT 9" }}{PARA 256 "" 0 "" {TEXT 286 0 "" }{TEXT 260 22 "GEOMETRIC APPLICATIONS" }}{PARA 256 "" 0 "" {TEXT 287 0 "" }{TEXT 261 14 "OF INTEGRATION" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 265 " The definite integral can be used to solve a varie ty of problems from geometry; examples are finding areas between curv es, lengths of curves, volumes and surface area of three dimensional \+ solids, and centroids of plane regions. In this laboratory we will use " }{TEXT 262 5 "MAPLE" }{TEXT -1 138 " to calculate the area between \+ curves and the volume of solids of revolution. The volume problem will also give us the opportunity to use " }{TEXT 263 5 "MAPLE" }{TEXT -1 25 "'s 3D graphing abilities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 19 "AREA BETWEEN CURVES" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " In class we showed th at if f (x) " }{TEXT 265 1 ">" }{TEXT -1 155 " g (x) for all x in [ a , b ] then the area A of the region enclosed by the curves y = f (x ), y = g (x), and the lines x = a a nd x = b is given by" }} {PARA 256 "" 0 "" {TEXT -1 4 "A = " }{XPPEDIT 18 0 "Int(f(x)-g(x),x = \+ a .. b);" "6#-%$IntG6$,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F*!\"\"/F*;%\"aG%\" bG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 105 "In practice, a la rge part of the problem in using this formula involves determining exa ctly where f (x) " }{TEXT 266 1 ">" }{TEXT -1 23 " g (x) and where g \+ (x) " }{TEXT 267 1 ">" }{TEXT -1 39 " f (x). We illustrate with an ex ample:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 268 7 "EXAMPLE" }}{PARA 0 "" 0 "" {TEXT -1 101 " Find the \+ total area of all regions enclosed by the curves \n f(x) = x + s in(2x) and g(x) = " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " The first thing to do is to plot the graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x->x+sin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$sinG6#,$*&\"\"#F .F-F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g:=x->x^ 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG% &arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "c:=textplot(\{[1.3,4,`g`],[3,1.5,`f`]\}):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "p:=plot(\{f,g\},-Pi..Pi,-5..5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(\{c,p\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%%TEXTG6$7$$\"#8!\"\"$\"\"%\"\"!Q\"g6\"-F$6$7$$\"\"$F, $\"#:F)Q\"fF.-%'CURVESG6$7\\o7$$!3*****4tk#fTJ!#<$!3G8/vMEfTJF>7$$!3w \"*pr)3PY+$F>$!3)=!=qo#QTt#F>7$$!3?3E[*ysa)GF>$!3ALw7$$!3gX&) )>2g9v#F>$!3yJUSp@*z/#F>7$$!3)4577.flh#F>$!3?*)e`[B5\\7$$!3u\"QM::* H#[#F>$!3`)))R#*oNR^\"F>7$$!3y\\AhcI#yN#F>$!3QGZ1h$GyN\"F>7$$!3/=ZVj\" zLH#F>$!3T%Qp+Kg7I\"F>7$$!3&e=d-FN*GAF>$!3;XAW:1:h7F>7$$!3IS]u.tGi@F>$ !3!GmsN]\\lB\"F>7$$!3u%*GBP$Rc4#F>$!3p,YJ\"Qs$G7F>7$$!3q4Yf6^?H?F>$!3x Ykl(\\NbB\"F>7$$!3mCj&f)3xi>F>$!3S)zwjZ*oc7F>7$$!3'3-$3>^V%*=F>$!3U\"f jTA=9H\"F>7$$!3(or4AN*4E=F>$!3@I%zuE$RP8F>7$$!3QQQ*3**=dq\"F>$!3EQjdDm 8R9F>7$$!3e!>jg@*>q:F>$!3;VFCPMRr:F>7$$!3`;m,%)H7M9F>$!3OZz3f$ySq\"F>7 $$!3v(4F,K))HI\"F>$!3J,[t*>fL\"=F>7$$!3Z)fe7!pWV7F>$!3ujhV\"HnB&=F>7$$ !3S*4!R#[0R=\"F>$!3m*eB:l]F)=F>7$$!3%H7'Qw\\586F>$!3d2xcs3!f!>F>7$$!3p Y@QqWIU5F>$!354c.39=8>F>7$$!3:o6z[:FB)*!#=$!3vQowuDw0>F>7$$!3Wp3w$R)\\ B#*F\\t$!3ervj\\A2&)=F>7$$!3N]#[4j>e_)F\\t$!3U-?96#oN%=F>7$$!3EJc8o39G yF\\t$!3!fqR>t+Gy\"F>7$$!3'=[BP-8If'F\\t$!3M&[$fd'owi\"F>7$$!3CB3<@?)y B&F\\t$!3OBcEqH+!R\"F>7$$!3_*)R7$$!38TqX=W2, EF\\t$!3%G*30%[S%3\"F>7 $$\"3*4*=Jof03_F\\t$\"3eG^@BV-%Q\"F>7$$\"3q]1XIqOClF\\t$\"3r\"))\\92(G <;F>7$$\"3%HoBlmlzz(F\\t$\"3udl`6Qtz7$$\"3ty;*ytA]])F\\t$\"3'zQ'[`r .U=F>7$$\"3^u'f#4)z?@*F\\t$\"3;))[!egXX)=F>7$$\"3%H[VGJ1>F>7$$\"39Hv$\"3(R.uY&4A8>F>7$$\"3DZ>^Mm5;6F>$\"3m)e(4A$ H_!>F>7$$\"3Olj%G%3%R=\"F>$\"3[PZZDat#)=F>7$$\"3\\-)REfwoI\"F>$\"3q#\\ )QLYa5=F>7$$\"3cM2vQyFT9F>$\"3C:'>\\hFup\"F>7$$\"3!y32s$*Qxc\"F>$\"3ww pFyO&Qd\"F>7$$\"35+K?Bs#**p\"F>$\"3;:S$[%o_W9F>7$$\"3<#H$eMa;H=F>$\"3s qK,u&=^L\"F>7$$\"3/UPl\\c\"o*=F>$\"3[h]\")pv+!H\"F>7$$\"3%>>CZ'eYk>F>$ \"3e\"\\@R)z)fD\"F>7$$\"3H>eDi5iH?F>$\"3x&yXmjXaB\"F>7$$\"3kYuyfix%4#F >$\"36k'z-&*p$G7F>7$$\"3-Gs$)foSh@F>$\"3]^dX-kLO7F>7$$\"3T4q))fu.GAF>$ \"3)f[<`Y12E\"F>7$$\"35:t#\\K;TH#F>$\"3LNlbEQ\"=I\"F>7$$\"3y?w'**=&>gB F>$\"3!*=U*4?F-O\"F>7$$\"3!*>3a=Wj\"[#F>$\"3P!=t=0SH^\"F>7$$\"3A?c*)yu \"3i#F>$\"3Ao/8q%Hwv\"F>7$$\"3-i-HnWIXFF>$\"37$$\"3u=RWhN. yGF>$\"3cvs&pw))\\P#F>7$$\"3ss(*RSA20IF>$\"3p^KW\"Q6at#F>7$$\"3!)***\\ /l#fTJF>$\"3t7/-%'COLOURG6&%$RGBG$\"*++++\"!\")$F,F,F_al-F86$7 S7$F<$!3]fiV\\wi+J!#;7$FB$!3k7%R=]RDr#Ffal7$FG$!3#p5[f9JCS#Ffal7$FL$!3 E(pytz,I3#Ffal7$FQ$!3'ztIl/'R\"z\"Ffal7$FV$!3F>7bW`aH:Ffal7$Fen$!35C52 V:z58Ffal7$F_o$!3N]$H*>!pt5\"Ffal7$Fio$!3^!3kb9GM?*F>7$Fcp$!3!HggK2::c (F>7$F]q$!3!zxQJ$=Q*3'F>7$Fbq$!3g)ppIk]F'\\F>7$Fgq$!3ARj&*HjOrQF>7$F\\ r$!3]C8$eEt&\\HF>7$Far$!34lK&pj&=7AF>7$F[s$!3C#f&\\N+Sf;F>7$Fes$!3>[`v E\"eB8\"F>7$F`t$!3e=VN^'*pYyF\\t7$Fjt$!3ev0&>Coqz%F\\t7$F_u$!3/yF#>3Re 'GF\\t7$Fdu$!32YJh1V.P9F\\t7$Fiu$!31#G%>!z%=^h!#>7$F^v$!3'QUn6#)z(f:\"F>7$Fez$\"3_kz*ps[&f;F>7$Fjz$\"3*eb872\\?B#F>7$F_[ l$\"3a2ORN-%R*HF>7$Fd[l$\"3_9J$o=(>`QF>7$Fi[l$\"3K;L=_!pB\"\\F>7$F^\\l $\"3?%eG[>1,7'F>7$Fh\\l$\"3=?IDd:7\"e(F>7$Fb]l$\"3#**paaoh?>*F>7$F\\^l $\"3+g=A@9.16Ffal7$Ff^l$\"3L@!))fw^ZJ\"Ffal7$F[_l$\"3&\\nRi)oJG:Ffal7$ F`_l$\"3!*\\b&fxc,!=Ffal7$Fe_l$\"3q'oM%*o_!p?Ffal7$Fj_l$\"3onl$=T(*QQ# Ffal7$F_`l$\"3zg%*)p@=Pr#Ffal7$Fd`l$\"3FEMtewi+JFfal-Fi`l6&F[al$\")Vyg >F^al$\")d@R!)F^alF[[m-%+AXESLABELSG6$Q!F.Fb[m-%%VIEWG6$;$!+aEfTJ!\"*$ \"+aEfTJFi[m;$!\"&F,$\"\"&F," 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" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Note that the curves intersect at the origin, since f(0) = g (0) = 0. Next, we try find the X-coordina tes of the other two points of intersection of the curves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(f(x)=g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%'RootOfG6#,(*&\"\")F&-%$sinG6# %#_ZGF&!\"\"*&\"\"%F&F1F&F2*$)F1\"\"$F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Apparently there is no \"nice\" formula for the solut ions, but we can still use \"fsolve\" to find decimal representations \+ for them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "X[1]:=fsolve(f (x)=g(x),x=-2..-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6#\"\"\" $!+ " 0 "" {MPLTEXT 1 0 33 "X[2]:=fsol ve(f(x)=g(x),x,x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"XG6#\" \"#$\"+" }{TEXT -1 50 " g(x), and the interval or intervals where g(x) " }{TEXT 270 1 ">" }{TEXT -1 45 " f(x).From the graph, one can see that g (x) " } {TEXT 271 1 ">" }{TEXT -1 18 " f(x) for x in [ " }{XPPEDIT 18 0 "X[1] " "6#&%\"XG6#\"\"\"" }{TEXT -1 21 ", 0 ] and that f (x) " }{TEXT 272 1 ">" }{TEXT -1 22 " g (x) for x in [ 0 , " }{XPPEDIT 18 0 "X[2]" "6#& %\"XG6#\"\"#" }{TEXT -1 3 " ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Finally, we set up the integrals and eval uate them:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "Area:=Int(g(x)-f(x),x=X[1]..0)+Int(f(x)-g(x),x=0..X [2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AreaG,&-%$IntG6$,(%\"xG!\" \"-%$sinG6#,$*&\"\"#\"\"\"F*F2F2F+*$)F*\"\"$F2F2/F*;$!+ " 0 " " {MPLTEXT 1 0 12 "value(Area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +;s.X@!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 31 "VOLUMES O F SOLIDS OF REVOLUTION" }}{PARA 0 "" 0 "" {TEXT -1 84 " A solid gen erated by revolving a curve y = f (x) about the X-axis is called a \+ " }{TEXT 274 19 "solid of revolution" }{TEXT -1 72 ". Examples of such solids are cylinders, cones and spheres. The command\n" }}{PARA 256 " " 0 "" {TEXT 291 57 "plot3d([x, f(x)*cos(t), f(x)*sin(t)], x=a..b, t=0 ..2*Pi);" }}{PARA 0 "" 0 "" {TEXT -1 58 "\nplots the surface generated by revolving y = f (x) , a " }{TEXT 275 1 "<" }{TEXT -1 3 " x " } {TEXT 276 1 "<" }{TEXT -1 50 " b about the X-axis. As an example, let f (x) = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 5 " , 0 \+ " }{TEXT 277 1 "<" }{TEXT -1 3 " x " }{TEXT 278 1 "<" }{TEXT -1 3 " 1. 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int(Pi*x^4,x=0..1); evalf(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"V G,$*&\"\"&!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3` =$G'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }