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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 16 "Double I
ntegrals" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 257 24 "Viewing the
 Construction" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Notes to Students:" }}{PARA 0 "" 
0 "" {TEXT -1 273 "This lab is designed for you to visually capture th
e essence of double integrals by the usage of animations. You should f
ocus on the connections between the single variable 2-D Riemann Rectan
gle Approximation Method and the new 3-D approach for functions of two
 variables. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
278 0 "" }{TEXT 274 0 "" }{TEXT 275 17 "Approximate time:" }{TEXT 276 
0 "" }{TEXT 277 11 " 30 minutes" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
13 "Introduction:" }}{PARA 0 "" 0 "" {TEXT -1 973 "Remember from Calcu
lus I that we wished to determine the area beneath the graph of a non-
negative function. This led to the Riemann Rectangle Method for approx
imating integrals. As we progressed we learned that the integral over \+
a finite interval produced a net accumulation (or signed area). We spe
nt a significant amount of time devoted to partitioning a region, util
izing various rectangle schemes (right-hand, left-hand, and midpoint),
 and then discovered the elegant Fundamental Theorem of Calculus appro
ach. In Calculus II we took our new found knowledge and determined the
 volumes of certain solids either by creating disks/washers or shells.
 We also determined volumes by summing up cross-sectional slices. Thro
ughout both of these two endeavors we used refinements of length and t
he notion of limits to produce the desired outcome. Double integrals a
re akin to such processes and are relatively easy to work with. The qu
estion we seek to address is outlined below:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 
118 "\"Given a function of two variables, say, f(x,y), that is continu
ous and non-negative on a region, R, in the xy-plane, " }{TEXT 261 15 
"find the volume" }{TEXT 262 80 " of the solid enclosed between the su
rface defined by f(x,y) and the region, R.\"" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "It is important to note the ana
logy to the area question from Calculus I." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 112 "\"Given a function of one va
riable, say, f(x), that is continuous and non-negative over a closed i
nterval [a,b], " }{TEXT 263 13 "find the area" }{TEXT -1 93 " enclosed
 beneath the function, f(x),  and above the horizontal axis on the int
erval [a, b].\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 211 "Our scheme of partitioning, limiting, and computing are \+
inherently identical. The following sections will explore the procedur
es in greater depth and should be used in conjunction to your text and
 lecture notes." }}{PARA 0 "" 0 "" {TEXT 260 1 " " }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 37 "The Double Integral over a Rectangle:" }}{PARA 0 
"" 0 "" {TEXT -1 284 "Multi-variable calculus deals with functions of \+
more than one variable. Such a case is the surface defined as f(x,y). \+
We note that for each ordered pair that is in f's domain, we obtain a \+
function output. The output is an ordered triple and is of the form (x
, y, f(x,y)) = (x, y, z). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 65 "Suppose we wish to determine the volume beneath a
 given surface, " }{TEXT 264 7 "f(x, y)" }{TEXT -1 55 ",  and the xy-p
lane over a closed rectangle defined as:" }}{PARA 0 "" 0 "" {TEXT -1 
53 "R = [a, b] x [c, d]. A sample surface is shown below:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{BITMAP 388 
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@;:::::::::::::::::::::::::::::::>:R>:::::::::::::::::::::::::::::::J:
ZK::::::::::::::::::::::::::::::::;:@;:::::::::::::::::::::::::::::::>
:R>:::::::::::::::::::::::::::::::J:ZK::::::::::::::::::::::::::::::::
;:@;:::::::::JB>Z:F\\:>:@J:>Z:><=J:<[KF\\:>ZK><R:;jB@[;>Z:JB><;B:@K:FZ
K>Z;>Z;F<;R>]B:R>]::::::::::J:ZK::::::::::B>@KBZJ>:;R:[B:]B>=B:@JB@JBF
<=J:>:]R>]B:]B>>Z;>:]ZK><<K:<;F\\;:]jB::::::::::>Z:R>@;:::::::::ZJJ:ZJ
:[jB:::]R>]R:]B>JBZK><<[JJ:R>;R:F<::::::::::>Z:R>]R>::::::::::<[K>:>ZJ
>:@JB:::F<]R:<;><R>[ZK>:;J:@ZK>Z;jB::::::::::J:<ZKF\\K::::::::::B><;F
\\:J:><]:::F<]Z::[:@KBR>J:R>>Z;F<]::::::::::J:F\\KR>::::::::::<[::=B>@
K:><:::]jB>\\:F:;R:[jBB:[B:<k:J:@JB@J:@J:F\\KF<::::::::::;B:@[K:::::::
:::B>>:<k:<K:B:[B:]B><:>\\;F\\:B>@J:@ZKF\\J>Z:F\\K>Z;>\\:F\\:B:[B:]B><
;R:>ZKR>]::::::::::J:<ZKR>::::::::::<;[B:@K:FZJ>Z;>Z;F<;B>@;;JBZJR>]R>
]B:B>@Z:><=Z:FZK>:=jBR><j:@[K::::::::::J:<:]R>::::::::::<[K><@[K:ZJ:]:
:B:=j:R>:[:@;::::::::::::;B:R>::::::::::<;[ZKR>:B>jB::<j:ZK:><[R>F<:::
:::::::::;B:R>::::::::::<[K>:@[K:ZJ:]::B:=B>=R>]R>:[JBjB::::::::::::>Z
:ZK::::::::::><;B:@:]::<:F<:ZJR:=ZK:B>@;]::::::::::::J:ZK:::::::::::::
:::::::::::::::::::;:@;:::::::::::::::::::::::::::::::>:R>::::::::::::
:::::::::::::::::::J:3:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 
" It is important to note the 2-D to 3-D analogy." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Volume Example 1
:(Animated)" }}{PARA 0 "" 0 "" {TEXT -1 54 "The following example show
s the surface defined as:   " }}{PARA 257 "" 0 "" {OLE 1 3596 1 "[xm]B
r=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyyqyyyYJ<j<Jyky;
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyj;j;tV;B:;JZC:bKB::>^Nxee
>[=_r;V:>Z@Z:j\\FHemj^HMmqnG;KaFFJufF>::::::;C:?jyyiy=J:B:::::::sT:B:F
:YLpfF>:::::::::J?N:ry:>:<::::::G:c:;:?ja:[LsfFaMR>`:J:<:::::::>=?R:?Z
:F:;jysy;Z::::::j=JNL:<:=:?J:VZ:J<j<>Z:nyyMyK>j>J?>:Q:S:UJ:n;v;;JBB:]:
_:a:c:e:gJ:v<nYvY:::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::s:?j>aWZ@J:F:<j?J:JZC:ER:LZ>WdG_dnR:arOMeU=DUSeJ=uVMuR
AtUCUS[TRIUSamBPJd`ppPps<LaPpu<Lcxpp@PqF\\=Vdsgg\\wgfS>ErJYUW_UTEeV;cM
EM:@K<pNd<l\\\\QtDqjN<KRBYUKYFKYSJ[Dw[r=ItQ=DN_yAwRJ[Dwer==D:cYKcYCcy]
?tQ=\\JrJyTJvk?_xdveB_xIHb:B><NtyrZxQ^LYBxjZdliLniPNtyo\\RJYB==R:=R:HJ
;N:?j:F:Aj:^:;b<[B;>Z:>w=Nbfgh_s==R:DZ=pF?J;]\\<>Z=@I?J;m\\<rZmE:?J;a
\\<^Z@hAuc:\\rRN:KfGDJ<N\\\\f_<rZkm;^=hj:@j:<:D=dJ;LJ=;K^<rZmm;::K^:B:
::::::F:;b:<Z::::::::::jysyA:C:=Z:vYxY;J:JsM;Qs:nw=j:F;HjuAZ:N:Q?;J<B:
?R:=j<j:Djyyyy=J<JQ@jFf:F`<FZ:B:EB:V[<Z:f:^[<>v<>ZMV<<JMTjA^=yyyxY:<Jv
>\\:B:;xyfoyyY;h><j@EZ:V[:>:G;Sj`@Pt\\Pd`QrP`:>FEZ:F[<jyEj<JDDJW@J\\pj
DjwEJfui:B:>l;B:<Z<B>cTTUUSZZ:W=C:US:N[:JSd:<J;>Z:N`lNw>>d<f:^<;c:KC<e
:qQ:tI<:[>;B:DZJ:::^:f?=J<JrA:SB:;B:_c<B:AB:;jRTMSR:og;Lj<>:c:kW:e:qAB
:^q<J:<Z<:::J<jP>:C:[q:^;N@B:A:KCEM[<b:Lj<JDJJg<VYZ:JXG:;B:;b:<Kd:::C:
Uk:^:>X;jA>Z:N@B:E:GEXmEXmM:>]<>gAN[?B;E:c:[h;[Z:VYZ:JBK:DJ:DZJVdscRYE
U@J<<:UK:^:>x;F:SZ:N`DZ:V:fagf?[w:Oy:LjOKMJ`@j`@PtZ_gdO?b;fZ;Z>WD:BU:R
S:^DP@:nG]jcfg\\wG@k<H>:>:j\\jjN<K:ZnA<I\\JOX>dni<<Z?;ZJrJ@>YVKd<cy]<Z
JBJFby?C[jZ:xo\\RJj:@Z=N:N:=j:V:=J<>ZD>\\>J:<Jn;F:DZ=pF<J;]<HRv<J;NjfZ
=Hi:NZ:NjcJ<fAB\\];KfG^:_:HRvV:sJ:hj:B:bF<ZDN::K^<rZmm;::K^:B:V[:>:G;S
jysy=J`>Z:>:[B:;B:qQBv:>:sO:B:=b:?bBaTXaEWEUU^:f?=J<Jr?:AJ:<J:::::::::
:3:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "over the rectangle
 [0,2] x [0,2]. The code will produce the approximate volume and the M
aple computed double integral. Place the cursor in the red " }{TEXT 
266 7 "restart" }{TEXT -1 53 " and press \"Enter\". Click on the pictu
re and use the " }{TEXT 267 17 "Animation Toolbar" }{TEXT -1 47 " at t
he top of the screen to run the animation." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 975 "restart:with(plot
s):with(plottools):\nf:=(x,y)->16-x^2-2*y^2:\nsurface:=plot3d(f(x,y),x
=0..2,y=0..2,style=wireframe,grid=[10,10],shading=none,color=blue,axes
=normal,thickness=2):\ndelx:=0.2:\ndely:=0.2:\nxnum:=ceil(2/delx):\nyn
um:=ceil(2/dely):\nxvals:=[seq(i*delx,i=0..xnum)]:\nyvals:=[seq(i*dely
,i=0..ynum)]:\ncolumns:=[seq(seq(cuboid([xvals[i],yvals[j],0],[xvals[i
+1],yvals[j+1],f(0.5*delx+xvals[i],0.5*dely+yvals[j])]),j=1..nops(yval
s)-1),i=1..nops(xvals)-1)]:\nframes:=[surface,seq(display(columns[1..4
*k],surface,view=[0..2,0..2,0..16]), k=1..nops(columns)/4),display(col
umns,surface,view=[0..2,0..2,0..16])]:\ndisplay(frames,insequence=true
,view=[0..2,0..2,0..16],title=\"Approximate Volume\",titlefont=[TIMES,
BOLD,14],labels=[\"x\",\"y\",\"z\"],orientation=[40,68],color=grey);\n
approximate_volume:=add(add(f(0.5*delx+xvals[i],0.5*dely+yvals[j])*del
x*dely,j=1..nops(yvals)-1), i=1..nops(xvals)-1);\ncomputed_volume:=Int
(Int(f(x,y),y=0..2),x=0..2)=evalf(int(int(f(x,y),y=0..2),x=0..2));\n \+
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 472 "Note the close agreement between the approximatin
g volume and the exact volume. Furthermore, we integrated with respect
 to y first. There is much more to be said about the order of integrat
ion and how the heights for the columns were determined but these ques
tions are best discussed in the classroom setting. Further examples of
 surfaces and their approximating volumes follow along with a student \+
problem set. Enjoy the examples and think about what each one is showi
ng." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 42 "                                          " }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 28 "Further Examples: (Animated)" }}{PARA 0 "" 0 "" 
{TEXT 268 10 "Example 2." }{TEXT 269 0 "" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The second example
 is the function:" }}{PARA 257 "" 0 "" {OLE 1 3592 1 "[xm]Br=WfoRrB:::
wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::fyyyyyqyyyYJ<j<Jyky;:::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::N
DYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyj;j;tV;B:;JZC:bKB::>^Nxee>[=_r;V:>Z@
Z:j\\FHemj^HMmqnG;KaFFJufF>::::::;C:?jyyiy=J:B:::::::cT:B:F:YLpfF>::::
:::::J?N:ry:>:<::::::G:c:;:?ja:[LsfFaMR>`:J:<:::::::>=?R:?Z:F:;jysy;Z:
:::::j=JGL:<:=:?J:VZ:J<j<>Z:nyyMyK>j>J?>:Q:S:UJ:n;v;;JBB:]:_:a:c:e:gJ:
v<nYvY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::s:?j>aWV@:=Z:V;>:>R<j<@Z>BKaTMcDG@jCXontpZlPpDnjHqnHp[xPlPPb@pmPp
sF\\?^dcgg_WhZnc_whZNdigg[oG]r:aTXUeRYEU@k<HnqPQs`plDQJtnl>ZK^ZG_dZfbr
_hlGF_J>@lqPnAMnQ@NbLYBxj]xoZ<OsyKI@NbLyDxjZ<JtQNtQLtyR[xoZB>Hni@>YVKc
iDyDLcy]=D:<[:_xIHby?cnA\\IFbdve>we?_xYgB@nAlj:@j:@Z=N:?J;F:=j;F:CJ:dj
:<JBL:;B:Ky:?DUeUSxj:DZ=pF?J;]\\<>Z=@I?J;m\\<rZmE:?J;a\\<^Z@hAuc:\\rVN
:DKKpL;Vjrs:l;QR:=R:;b:\\RSJ?DJ<TBjvV:::B:xxqyH<:::::::F:;b:<Z::::::::
::jywyyY:^:F:<jysy?:;:=i>V_=JjIZ:F:MZ=>x;B:?jMK:^:NZ;F:E:=b:yyyyI:C:WS
:mj<jRDj:<Z:fZ:j?D:;B:E:Sb:;i:;S>aZ:n^@v;sjyyiyAZ:>I[B:<JZyIUwyyQj=;Uf
:<j?<:G;Sj`@Pt\\Pd`QrP`:>FEZ:F[<jyEj<JDDJW@JpljDjwEJfui:B:>l;Z:b:;B>cT
TUUSZZ:W=CJ:f_;J?<:_c<B:?J:<JSDMkIJbDj<JD>^<>dBf<VY;byB:>L>Z:b:<;::J<j
PF:C:[Y:F[:>Z:N`DZ:FZ:B:m[>f:;JDJVAjDjw;<:sg:B:D::::CB:f?;J<JrG:SJ:<JS
Z:VZ:JNllNb:DZ>f:^<>Ne:qAB:^Q=J:<J:DZJ^D::J<jPF:C:[Q:v;;B:_;<j<Jm\\qv
\\qv>:kb:[f;?B;Lj<JDJbUJB<Z:VYZ:JBA:;b:;b:<kc`a\\wgfS:CB:f?;J<JrAj:^;<
JSd:<j;jXplP>u<Nw=B;I?G;CT:UTRc=PpdPOV@j<@:Lnc:X?l@::JdRS:JqF<alpDpqR>
ErJ:J::E<=MC>;Y=Bw[j]BN_A[DweZ:RK:B>H^;kAQ^dJtyb:B><>=tYKLBFBZYgB@>FZ;
r:?:?j:F:Aj:^:;b<=B:L:;B:KAj:rZG]:NjbJ:HRv<J;NjfZ=Hi:NZ:NjcJ<fAB\\m;D;
Vjrs:l[:V[;J:DZBP@;J?D::mm;::Z:ruYq?JMJ@vYyyy=J`>Z:>:[B:;B:qQBv:>:sO:B
:=b:?bBaTXaEWEUU^:f?=J<Jr?:AJ:<J::::::::::::1:" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "on the rectangle [0,3] x [0,3]." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "What figure an
d approximating value would you expect the code to produce? Run the co
de and compare it with your conjecture.Place the cursor in the red " }
{TEXT 273 7 "restart" }{TEXT -1 19 " and press \"Enter\"." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 956 "rest
art:with(plots):with(plottools):\nf:=(x,y)->9-x^2-y^2:\nsurface:=plot3
d(f(x,y),x=0..3,y=0..3,style=wireframe,grid=[10,10],shading=zhue,axes=
normal,thickness=2):\ndelx:=0.3:\ndely:=0.3:\nxnum:=ceil(3/delx):\nynu
m:=ceil(3/dely):\nxvals:=[seq(i*delx,i=0..xnum)]:\nyvals:=[seq(i*dely,
i=0..ynum)]:\ncolumns:=[seq(seq(cuboid([xvals[i],yvals[j],0],[xvals[i+
1],yvals[j+1],f(0.5*delx+xvals[i],0.5*dely+yvals[j])]),j=1..nops(yvals
)-1),i=1..nops(xvals)-1)]:\nframes:=[surface,seq(display(columns[1..4*
k],surface,view=[0..3,0..3,0..9]), k=1..nops(columns)/4),display(colum
ns,surface,view=[0..3,0..3,0..9])]:\ndisplay(frames,insequence=true,vi
ew=[0..3,0..3,0..9],title=\"Approximate Volume\",titlefont=[TIMES,BOLD
,14],labels=[\"x\",\"y\",\"z\"],orientation=[40,68],color=grey);\nappr
oximate_volume:=add(add(f(0.5*delx+xvals[i],0.5*dely+yvals[j])*delx*de
ly,j=1..nops(yvals)-1), i=1..nops(xvals)-1);\ncomputed_volume:=Int(Int
(f(x,y),y=0..3),x=0..3)=evalf(int(int(f(x,y),y=0..3),x=0..3));\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
58 "Are the results consistent with what you previously know? " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 47 "Can you determine why this is an overestimate? \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 0 "" }{TEXT 271 
10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT -1 150 "This last example take
s a bit of a more complicated figure and shows the same visual dynamic
s for approximating the volume. The surface is defined as:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {OLE 1 5128 1 "[xm]Br=WfoRrB
:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:K:wAyA:::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JJLsihL>HN\\A
j;JZ`:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>:::::
:::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<ja:MLqnFYM:>::::::::N<<jyyiy=J:B:::
:::V:;J<>:;:wyyN::wyyyq<J=j=B:K:M:OJ:nY^;f;;JAjA>:[Z:F<N<;jCJDjDJE>:i:
k:m:o:q:s:u:w:y:;C:<jJ<:?;A;C;wAyA::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::=Z<Z::::::::::j:nyyYZDjysyAbm?:;JZC:bKg:UTRcET
cTX[US;CGWCF;SK]UW=EWMuUWM@B::P`?oBY:ETV:WCKaTMcDGLyVDurB:;::::::A:;::
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j:F:Aj:^:;b<=B:[B;>Z:>w=Nbfgh_s==b:HRmN:?FDDZ=@i:N:?FEElrfH=MtFGYMq>>W
lj:gmlJ::::::>^:N:yay=J:B::::::^:^o;>:;B:N:YLpJbNHEms>@[C:>Z::::::::kJ
;@j;>:C:yayA:<::::::Q:kd;B::::::::::::jysy:>:<:::::::::::::::::::vYxI:
;Z::::::::Z<rZmE:?J;a\\<^Z@hAuc:\\RSNZ<Nj]c:;jqN:Ha=R:Aj:<j;V;@Z<B\\]C
:@:N@;B:Aj:H:;Z=@IAJHr<=R:=B:?J>[DNZ>njerD?j;aa=::J:>ZDB:gFDDJ:TsNJ;BD
jNb:C:B\\eK:<J::::::yayYZ:J<j:>Z:vYxY;J:JUp;gd;<JIT:=j>r:Yr:^:W@;J<>:?
R:=j<>:=b:yyyyI:C:WS:mj<jRDj:<:E:Qb:<Z:f:^[<>r?>jTV<<JMTjA^=yyyxY:<Jv>
\\:B:;xyVnyyyyAbTjkP:QB:<JMJ@fc[KdRS;B:[S:MZ:Ny<B:m:;jyEJK^:f_;j<B:cb:
;i:AV=EZ:V\\<>v<VpHf:F[<jyg:^<Mw:qi=[J:<jwEj^myQ@:;B:[V:<Z<>Z<>ZJVdsgG
l`;^Z:B:Uk:F;;JSd:;B:=B:>?f:^\\<JuQKB<J:<jw;B:[V:b:DZJ:Y=<:UK;^:>x;F:>
:_c<B:=:MK:f:^\\<Ni<V]Kf<VYkhy<Z:JBA:<Z<Jd``_;<<nG^:f?=J<JrOZ:N;N@B:?J
:<JSDMoIJbDj<JDf\\<VmGf<VY;byB:>LB:DZJ:::^:f??J<JrG:MJ:N`DZ:F:F`:>Z>fZ
:>:cJjHJj\\jDjw;<:[V:>Z:b::::^Z:jPF:C:[Q;N;N@B:?J:N`oNyJB;E:cJvMjKqjDj
w;<:[><Z<B>:::CB:f??J<JrG:MJ:N`DZ:F:N`:>Z>f:^<;MVAjDjw;<:sg:>Z:b::::^Z
:jPF:C:[Q;>\\:B:_c<B:G:KCEMsRYuVDZ<>u=>V@Z>f:^<>n:f<VYZ:JXG:>Z<B>c<::^
:f??J<JrG:YJ:N@B:E:GEXmEXmE:;b:kb:ys<OB;Lj<JDf`<>hF><;B:qAB:>l;B:DJ:DJ
:<kcbRY=<:Uk:^Z:JrAJ;N;N`DZ:N:;JmUYgdZ>f:^\\<>F[h;SJ:<JSZ:VZ:>:usDQkrh
jNQZ>f:^<my:[E=O:_;<J;jmYYgd:JDVa?>hFN;N@B:?:KIxod<nSn>Nl;fc[_HRSeTOGV
:ER:BKa<@:BUFF:@@JdRS:JqF<alpDpqR>ErJY=:;:fBFF_J>vGZnA<I\\JOX>dni<<JtQ
>:<[=SJvk?cD^xID:<[:_X=tYKLBxjZ:xo\\RJj:@Z=NZ:N:?j:F:Aj:<J<>ZDFZ::;B:K
y:ZIF:HRmJ;]\\<>Z=@i:J;mL:DZ=Hi:NZ:NjcJ<fAJSNZ<:;jqZm;AB:=B:Aj?R:DZBH@
@:N@;j;FZ=J:rZkm;<JH>ZEF:<J;>Kb<?B;h<Vjss:::>:;b<<JE]<TsNB:?Zb:M[<^:ZB
h@;j?<Z:n>^;yayI:SL:<J:F[:>Z:NY<Z::N^:^:f??JB<J:<jw?<I:;JX?:<j:DJ;Dlc`
qsLqlp@C:Uk;^:>X=j;>Z:>::::::::::::::::::::::::::::::::::::::::::::1:
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 135 "over the same rectangle, namely [0,3] x [0,3]. Consider \+
how the cosine function should affect the surface. Place the cursor in
 the red " }{TEXT 272 7 "restart" }{TEXT -1 19 " and press \"Enter\".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
963 "restart:with(plots):with(plottools):\nf:=(x,y)->cos(0.5*(x^2+y^2)
)+5:\nsurface:=plot3d(f(x,y),x=0..3,y=0..3,style=patch,grid=[10,10],sh
ading=zhue,axes=normal,thickness=2):\ndelx:=0.3:\ndely:=0.3:\nxnum:=ce
il(3/delx):\nynum:=ceil(3/dely):\nxvals:=[seq(i*delx,i=0..xnum)]:\nyva
ls:=[seq(i*dely,i=0..ynum)]:\ncolumns:=[seq(seq(cuboid([xvals[i],yvals
[j],0],[xvals[i+1],yvals[j+1],f(0.5*delx+xvals[i],0.5*dely+yvals[j])])
,j=1..nops(yvals)-1),i=1..nops(xvals)-1)]:\nframes:=[surface,seq(displ
ay(columns[1..4*k],surface,view=[0..3,0..3,0..9]), k=1..nops(columns)/
4),display(columns,surface,view=[0..3,0..3,0..9])]:\ndisplay(frames,in
sequence=true,view=[0..3,0..3,0..9],title=\"Approximate Volume\",title
font=[TIMES,BOLD,14],labels=[\"x\",\"y\",\"z\"],orientation=[40,68],co
lor=grey);\napproximate_volume:=add(add(f(0.5*delx+xvals[i],0.5*dely+y
vals[j])*delx*dely,j=1..nops(yvals)-1), i=1..nops(xvals)-1);\ncomputed
_volume:=Int(Int(f(x,y),y=0..3),x=0..3)=evalf(int(int(f(x,y),y=0..3),x
=0..3));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 39 "  Procede to the Student Problem Set.  " }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 20 "Student Problem Set:" }}{PARA 0 "" 0 "
" {TEXT -1 98 "There are 5 surfaces listed below. Use the code that ap
pears at the end of the list adjusting the " }{TEXT 279 15 "/*Function
 Line" }{TEXT -1 130 " to produce each surface and its approximate vol
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-1 10 "Surface 1:" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 
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start:with(plots):with(plottools):\nf:=(x,y)->place your function here
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{MPLTEXT 1 0 895 "\nsurface:=plot3d(f(x,y),x=0..3,y=0..3,style=patch,g
rid=[10,10],shading=zhue,axes=normal,thickness=2):\ndelx:=0.3:\ndely:=
0.3:\nxnum:=ceil(3/delx):\nynum:=ceil(3/dely):\nxvals:=[seq(i*delx,i=0
..xnum)]:\nyvals:=[seq(i*dely,i=0..ynum)]:\ncolumns:=[seq(seq(cuboid([
xvals[i],yvals[j],0],[xvals[i+1],yvals[j+1],f(0.5*delx+xvals[i],0.5*de
ly+yvals[j])]),j=1..nops(yvals)-1),i=1..nops(xvals)-1)]:\nframes:=[sur
face,seq(display(columns[1..4*k],surface,view=[0..3,0..3,0..9]), k=1..
nops(columns)/4),display(columns,surface,view=[0..3,0..3,0..9])]:\ndis
play(frames,insequence=true,view=[0..3,0..3,0..9],title=\"Approximate \+
Volume\",titlefont=[TIMES,BOLD,14],labels=[\"x\",\"y\",\"z\"],orientat
ion=[40,68],color=grey);\napproximate_volume:=add(add(f(0.5*delx+xvals
[i],0.5*dely+yvals[j])*delx*dely,j=1..nops(yvals)-1), i=1..nops(xvals)
-1);\ncomputed_volume:=Int(Int(f(x,y),y=0..3),x=0..3)=evalf(int(int(f(
x,y),y=0..3),x=0..3));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
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