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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 13 "A Witch's Hat" }}{PARA 
19 "" 0 "" {TEXT -1 23 "by Jules Guay , Teacher" }}{PARA 256 "" 0 "" 
{TEXT -1 38 "Cegep of Drummondville, Quebec, Canada" }}{PARA 257 "" 0 
"" {URLLINK 17 "Email: jules.guay@dr.cgocable.ca" 4 "" "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The goals \+
of the example presented in this worksheet are :" }}{PARA 0 "" 0 "" 
{TEXT -1 352 "     - To help the students to understand that a 3D plot
 of a solid of revolution can be acheived in different ways:.\n       \+
(spacecurve Command , parametric  form or using tubeplot Command )\n  \+
   - To help students to visualize how a volume or an area of revoluti
on is built-up\"\n        (with disks or cylinder shells for volume an
d rings for area ) ." }}{PARA 0 "" 0 "" {TEXT -1 80 "     - To demonst
rate the power of Maple to approximate the result of integrals\"" }}
{PARA 0 "" 0 "" {TEXT -1 115 "         (with evalf Command ) or with a
 Riemann Sum using many terms when the integral cannot be solved exact
ly . " }}{PARA 0 "" 0 "" {TEXT -1 22 "                      " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Packages used :  plots   plottools
 (line , rotate )  and  student ( middlesum) ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "NOTE: It can tak
e a few minutes to load the full worksheet depending  the PC used beca
use many 3D plot structures and animations are presented  ... it is wo
rth it ...  please be patient ... Thank You!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Plotting with  spacecur
ve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
"with(plots); with(plottools,rotate);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "cur:=spacecurve([0,t,exp(t)],t=0..3,color=khaki);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Curv:=seq(rotate(cur,k*Pi/3
6,[[0,3,0],[0,3,exp(3)]]),k=1..72):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 120 "display(Curv,axes=normal,labels=[x,y,z],orientation=
[4,84],title=\" drawing the witch hat\",titlefont=[COURIER,BOLD,12]);
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Plotting the curve in para
metric form with the drawing of an element of volume ( a disk fom the \+
rotation of a rectangle building a part of the volume  ) ." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "restart:with(plots):\nf:=exp(y)-1;
\nF:=unapply(f,y);\nrot:=3:\nnImage:=1:\nfor n from 1 to 64 \n   do \n
     \n      GrC:=plot3d([(y-rot)*sin(t),(y-rot)*cos(t)+rot,f],\n     \+
     y=0..3,t=0..2*Pi,color=khaki):\n      GrA:=pointplot3d([[0,rot,-1
],[0,rot,F(rot)+0.5]],\n             style=line,color=blue):\n      rd
isq:=seq(pointplot3d([[0,rot,F(1.5)],\n          [(1.5-rot)*sin(k*Pi/3
2),\n          (1.5-rot)*cos(k*Pi/32)+rot,F(1.5)]],\n          style=l
ine,thickness=8,color=maroon),k=1..n):\n      gr.nImage:=display(GrC,G
rA,rdisq):\n       nImage:=nImage+1:\n    od:\n\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 176 "display(gr.(1..nImage-1),insequence=true,st
yle=wireframe,labels=[x,y,z],orientation=[6,70],axes=framed,title=\"An
imation : Rectangle in rotation  \",titlefont=[COURIER,BOLD,12]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Now using tubeplot to draw the 3D \+
curve and filling up the witch hat." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "restart:\nwith(plots):wit
h(plottools):\nrad1:=y->ln(y+1);\nlst:=NULL:\nNimage:=1:\nparam1:=colo
r=magenta:\nparam2:=color=black:\nfor k from 1 to 19\n   do\n     if i
rem(k,2)=0\n       then\n           param:=param1\n        else\n     \+
      param:=param2\n      fi:\nCur:=tubeplot([3,y,0,y=0..exp(3)-1],ra
dius=3-rad1(y),color=khaki,style=wireframe,tubepoints=24):\nAX:=pointp
lot3d([[3,-2,0],[3,exp(3)+2,0]],style=line,color=blue,thickness=2):\n
\nL:=line([3,(k-1),0],[ln((k-1)+1),k-1,0],param,thickness=9):\nrdis:=s
eq(rotate(L,n*Pi/32,[[3,-1,0],[3,exp(3),0]]),n=0..64):\nlst:=lst,rdis:
\ngr.Nimage:=display(Cur,AX,lst):\nNimage:=Nimage+1:\nod:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "display(gr.(1..Nimage-1),in
sequence=true,orientation=[-90,-6],title=`Animation : Filling up the h
at`,titlefont=[COURIER,BOLD,12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
24 "Evaluating the volume : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "dV:=Pi*(3-ln(y+1))^2:\nVolum
e:=Int(dV,y=0..exp(3)-1)=int(dV,y=0..exp(3)-1)*`   cubic units `;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 47 "Filling up the witch hat with cylinder shells ." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 811 "
restart:\nwith(plots):with(plottools):\nrad1:=y->ln(y+1):\nparam1:=col
or=yellow:param2:=color=magenta:\nNimage:=1:\nlst:=NULL:\nfor k from 1
 to 20 \n   do\n     if irem(k,2)=0\n       then\n          param:=par
am2\n       else\n          param:=param1\n     fi: \nGR1:=tubeplot([3
,y,0,y=0..exp(3)-1],radius=3-rad1(y),color=khaki,style=wireframe,axes=
normal,orientation=[-90,10],tubepoints=30):\nGR2:=tubeplot([3,y,0,y=-2
..22],radius=3-2.98,color=black):\nCyDv:=tubeplot([3,y,0,y=0..exp(.15*
k)-1],radius=3-(.15*k),param,thickness=4,axes=normal,orientation=[-90,
10],tubepoints=30,style=wireframe):\nlst:=lst,CyDv:\ngr.Nimage:=displa
y(GR1,GR2,lst):\nNimage:=Nimage+1:\n   od:\n\ndisplay(gr.(1..Nimage-1)
,insequence=true,axes=none,orientation=[45,-1],title=\" Animation : Fi
lling up the hat \" ,titlefont=[COURIER,BOLD,12],labels=[x,z,y]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "dV:=2*Pi*(3-x)*(exp(x)-1):\n
Volume:=Int(dV,x=0..3)=int(dV,x=0..3)*`     cubic units`;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The integ
ral can be evaluate exactly ( both results for the volume are equal ).
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Looking for t
he area of the witch hat . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 704 "restart:\nwith(plots):\nset
options3d(axes=none,tubepoints=30,style=wireframe):\nrad1:=y->ln(y+1);
\nparam1:=color=cyan:\nparam2:=color=magenta:\nNimage:=1:\nlst:=NULL:
\nfor k from 1 to 20 \n   do\n     if irem(k,2)=0\n       then\n      \+
    param:=param1:\n       else\n          param:=param2:\n     fi:\nG
R1:=tubeplot([3,y,0,y=0..exp(3)-1],radius=3-rad1(y),color=khaki):\nGR2
:=tubeplot([3,y,0,y=-2..exp(3)+2],radius=3-2.98,color=black):\nGR3:=tu
beplot([3,y,0,y=((k-1)/20)*(exp(3)-1)..(k/20)*(exp(3)-1)],radius=3-rad
1(y),param):\nlst:=lst,GR3:\ngr.Nimage:=display(GR1,GR2,lst):\nNimage:
=Nimage+1:\n    od:\ndisplay(gr.(1..Nimage-1),insequence=true,orientat
ion=[-92,-21],title=`Animation : Area `,titlefont=[COURIER,BOLD,12]);
\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Evaluation of the area ..
........" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "restart:\nC:=exp(x)-1;\nX:=solve(C=y,x);\ndA:=2*Pi*(
3-X)*sqrt(1+diff(X,y)^2);\nA:=int(dA,y=0..exp(3)-1);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 125 "Don't worry Maple can proceed numerically to g
ive a good approximation . ( or with a Riemann Sum with many...many  t
erms .) ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "AreaApp:=evalf
(A,7)*`  square units`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 
"with(student):Digits:=18:\ndArea:=unapply(dA,y):\nA_Riemann:=middlesu
m(dArea(y),y=0..exp(3)-1,2000);\nRiemannApprox:=evalf(A_Riemann,7)*`  \+
square units`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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