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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 29 "Lesson 3  \+
Volumes using Disks" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 92 "Here are the solids of revolution for the
 examples which were worked in class using disks.  " }{TEXT 314 76 "Pr
ess the ENTER key on each line of command in order to execute the comm
and." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 9 "Example 1" }{TEXT 259 
0 "" }{TEXT -1 4 "    " }{TEXT 260 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "3/x;" "6#*&\"\"$\"\"\"%\"xG!\"\"" }{TEXT -1 4 "  , " }{TEXT 262 
1 "x" }{TEXT -1 6 " = 2, " }{TEXT 263 1 "x" }{TEXT -1 6 " = 6, " }
{TEXT 264 1 "y" }{TEXT -1 25 " = 0 ; revolve about the " }{TEXT 265 1 
"x" }{TEXT -1 5 "-axis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "tubeplot([x,0,0], x=2..6, radius=3/
x, tubepoints=40, color=pink);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }{TEXT 267 9 "Exampl
e 2" }{TEXT 268 0 "" }{TEXT -1 3 "   " }{TEXT 269 1 "y" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "sqrt(-x);" "6#-%%sqrtG6#,$%\"xG!\"\"" }{TEXT -1 4 
" ,  " }{TEXT 270 1 "y" }{TEXT -1 7 " = 0 , " }{TEXT 271 1 "x" }{TEXT 
-1 27 " = - 3 ; revolve about the " }{TEXT 272 1 "x" }{TEXT -1 5 "-axi
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "tubeplot([x,0,0], x=-3..0, radius=sqrt(-x), tubepoint
s=40, color=pink, orientation=[-107,82]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 0 "" }{TEXT 
274 9 "Example 3" }{TEXT 275 0 "" }{TEXT -1 3 "   " }{TEXT 276 1 "y" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 5 " \+
 ,  " }{TEXT 277 1 "y" }{TEXT -1 7 " = 4 , " }{TEXT 278 1 "x" }{TEXT 
-1 25 " = 0 ; revolve about the " }{TEXT 279 1 "y" }{TEXT -1 5 "-axis
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "tubeplot([0,0,z], z=0..4, radius=sqrt(z), tubepoints=
40, color=pink);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 280 0 "" }{TEXT 281 9 "Example 4" }{TEXT 
282 0 "" }{TEXT -1 3 "   " }{TEXT 283 1 "y" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "1/x;" "6#*&\"\"\"\"\"\"%\"xG!\"\"" }{TEXT -1 5 "  ,  " 
}{TEXT 284 1 "y" }{TEXT -1 7 " = 1 , " }{TEXT 285 1 "y" }{TEXT -1 7 " \+
= 4 , " }{TEXT 286 1 "x" }{TEXT -1 25 " = 0 ; revolve about the " }
{TEXT 287 1 "y" }{TEXT -1 5 "-axis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "tubeplot([0,0,z], z=1..4, r
adius=1/z, tubepoints=40, color=pink);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 288 0 "" }{TEXT 
289 9 "Example 5" }{TEXT 290 0 "" }{TEXT -1 4 "    " }{XPPEDIT 18 0 "y
 = x^2-6*x;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"*&\"\"'F)F'F)!\"\"" }{TEXT 
-1 4 " ,  " }{TEXT 291 1 "y" }{TEXT -1 17 " = 0 ; about the " }{TEXT 
292 1 "x" }{TEXT -1 5 "-axis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "tubeplot([x,0,0], x=0..6, ra
dius=6*x-x^2, tubepoints=40, color=pink);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Below is the solid cut in
 half in order to see inside it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "tubeplot([x,0,0], x=0..3, ra
dius=6*x-x^2, tubepoints=40, color=pink, orientation=[44,45]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "He
re are the solids of revolution for the examples which were worked in \+
class using washers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 0 "" }{TEXT 294 0 "" }{TEXT 295 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 0 "" }{TEXT 297 9 "Exampl
e 1" }{TEXT -1 3 "   " }{XPPEDIT 18 0 "y = 2*x^2;" "6#/%\"yG*&\"\"#\"
\"\"*$%\"xG\"\"#F'" }{TEXT -1 5 "  ,  " }{XPPEDIT 18 0 "y = 9-x^2;" "6
#/%\"yG,&\"\"*\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 22 "  ; revolve about
 the " }{TEXT 298 1 "x" }{TEXT -1 5 "-axis" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "tubeplot([x,0,0],
 x=-sqrt(3)..sqrt(3), radius=2*x^2, tubepoints=40, color=pink): solid1
1 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "tubeplot([x,0,0]
, x=-sqrt(3)..sqrt(3), radius=9-x^2, tubepoints=40, color=pink): solid
12 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(solid11
,solid12);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "Below is the solid cut in half in order to see inside it:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "tubeplot([x,0,0], x=-sqrt(3)..0, radius=2*x^2, tubepo
ints=40, color=pink): solid13 := %:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "tubeplot([x,0,0], x=-sqrt(3)..0, radius=9-x^2, tubepo
ints=40, color=pink): solid14 := %:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "display(solid13,solid14);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 299 0 "" }{TEXT 
300 9 "Example 2" }{TEXT 301 0 "" }{TEXT -1 3 "   " }{XPPEDIT 18 0 "x \+
= -y^2;" "6#/%\"xG,$*$%\"yG\"\"#!\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 
18 0 "y-x-2 = 0;" "6#/,(%\"yG\"\"\"%\"xG!\"\"\"\"#F(\"\"!" }{TEXT -1 
21 " ; revolve about the " }{TEXT 302 1 "y" }{TEXT -1 5 "-axis" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 80 "tubeplot([0,0,z], z=-2..1, radius=2-z, tubepoints=40, color=pink
): solid21 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "tubeplo
t([0,0,z], z=-2..1, radius=z^2, tubepoints=40, color=pink): solid22 :=
 %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(solid21,soli
d22);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "Below is the top of the solid cut through the " }{TEXT 
303 1 "x" }{TEXT -1 70 "-axis in order to see inside it.  The view is \+
looking upward from the " }{TEXT 304 1 "x" }{TEXT -1 6 "-axis:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 102 "tubeplot([0,0,z], z=0..1, radius=2-z, tubepoints=40, color=pink
, orientation=[117,153]): solid23 := %:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "tubeplot([0,0,z], z=0..1, radius=z^2, tubepoints=40,
 color=pink, orientation=[117,153]): solid24 := %:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "display(solid23,solid24);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Below is the bo
ttom of the solid cut through the " }{TEXT 307 1 "x" }{TEXT -1 72 "-ax
is in order to see inside it.  The view is looking downward from the \+
" }{TEXT 305 0 "" }{TEXT -1 0 "" }{TEXT 306 1 "x" }{TEXT -1 6 "-axis:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "tubeplot([0,0,z], z=-2..0, radius=2-z, tubepoints=30
, color=pink, orientation=[47,47]): solid25 := %:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 100 "tubeplot([0,0,z], z=-2..0, radius=z^2, tube
points=30, color=pink,orientation=[47,47]): solid26 := %:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(solid25,solid26);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 308 0 "" }{TEXT 309 9 "Example 3" }{TEXT 310 0 "" }{TEXT -1 3 " \+
  " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 
4 " ,  " }{TEXT 311 1 "y" }{TEXT -1 8 " = 0 ,  " }{TEXT 312 1 "x" }
{TEXT -1 25 " = 9 ; revolve about the " }{TEXT 313 1 "y" }{TEXT -1 5 "
-axis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "tubeplot([0,0,z], z=0..3, radius=z^2, tubepoints=40, \+
color=pink): solid31 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
77 "tubeplot([0,0,z], z=0..3, radius=9, tubepoints=40, color=pink): so
lid32 := %:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 35 "The view of the solid from the top:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(sol
id31,solid32);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 38 "The view of the solid from the bottom:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "tub
eplot([0,0,z], z=0..3, radius=z^2, tubepoints=40, color=pink, orientat
ion=[46,140]): solid33 := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 99 "tubeplot([0,0,z], z=0..3, radius=9, tubepoints=40, color=pink, o
rientation=[46,140]): solid34 := %:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "display(solid33,solid34);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }