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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 1
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Define a function " }{XPPEDIT 
18 0 "g;" "6#%\"gG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "g(x) = sqrt(9-x
^2);" "6#/-%\"gG6#%\"xG-%%sqrtG6#,&\"\"*\"\"\"*$F'\"\"#!\"\"" }{TEXT 
-1 48 " .  What surface is generated when the graph of " }{XPPEDIT 18 
0 "g;" "6#%\"gG" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "x = -3;" "6#/
%\"xG,$\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 3;" "6#/%\"
xG\"\"$" }{TEXT -1 24 " is revolved around the " }{XPPEDIT 18 0 "x;" "
6#%\"xG" }{TEXT -1 37 "-axis?  Confirm your answer by using " }{TEXT 
256 6 "plot3d" }{TEXT -1 68 " to draw the surface.  (Hint: cut-and-pas
te from the example above!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 9 "Solution." }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g := x-> sqrt(9 - x^2) ;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowG
F(-%%sqrtG6#,&\"\"*\"\"\"*$)9$\"\"#F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y = sqrt(9-x^2);" "6#/
%\"yG-%%sqrtG6#,&\"\"*\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 33 " is the u
pper half of the circle " }{XPPEDIT 18 0 "x^2+y^2 = 9;" "6#/,&*$%\"xG
\"\"#\"\"\"*$%\"yGF'F(\"\"*" }{TEXT -1 93 ", so it is a semi-circle of
 radius 3, centred at the origin.  When it is revolved around the " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 42 "-axis, we should get a sphe
re of radius 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d([
x,g(x)*cos(t), g(x)*sin(t)], x=-3..3, t=0..2*Pi, axes=boxed, lightmode
l=light1);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "How should we proceed in order to
 find the area of such a surface?  Naturally, we begin by dividing the
 interval " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 305 " \+
over which we are working into a certain number of subintervals, and f
inding an approximation to the area of the part of the surface corresp
onding to one subinterval.  This is the most complicated approximation
 we have seen so far, and it cannot be done in one step, but the first
 step is to approximate " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 188 
" on each subinterval by a line segment, exactly as we did when we com
puted arclengths.  Here is the procedure from the arclength worksheet \+
that returns a piecewise-linear approximation to " }{XPPEDIT 18 0 "f;
" "6#%\"fG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "pl := proc(f,a,b,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "        \+
local u,v,k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "        k := 1 + fl
oor(n*(x-a)/(b-a));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "        u :=
 a + (k-1)*(b-a)/n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "        v :=
 a + k*(b-a)/n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "        unapply(
f(u) + (f(v) - f(u))/(v-u)*(x-u), x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "      end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Working with \+
the function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 78 " we defined
 above, we can plot the piecewise-linear approximations, and (with " }
{TEXT 257 6 "plot3d" }{TEXT -1 56 ") the surfaces we get by revolving \+
these approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pl
ot([f,pl(f,-1,2,5)], -1..2, thickness=2);" }}{PARA 13 "" 1 "" 
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s$\"35+]A,/L/BF37$Fis$\"3'***\\(*)*QF'[#F37$F^t$\"3Q++0hyx#o#F37$Fct$
\"3s++!=o&y&)GF37$Fht$\"30***\\+9*\\1JF37$F]u$\"3P**\\i1KT:MF37$Fbu$\"
3c++:4'4%ePF37$Fgu$\"3#z***>KrgmSF37$F\\v$\"3E++]rPn&R%F37$Fav$\"3?**
\\ibq&Qp%F37$Ffv$\"3K,+qX,&)>]F37$F[w$\"3F**\\AQUeE`F37$F`w$\"3=+]KY(4
sk&F37$Few$\"3+++]&yz1'fF37$Fjw$\"3k**\\PfEuziF37$F_x$\"3#4++;$3)3e'F3
7$Fdx$\"3m++02\"Q)))oF37$Fix$\"3a,]xS`C%>(F37$F^y$\"3)4++-(G)[Z(F37$Fc
y$\"3q++&)[^_'z(F37$Fhy$\"3u++?Pe?%3)F37$F]z$\"3;+]<`M$4R)F37$Fbz$\"3S
,]A>4^%o)F3Ffz-F\\[l6&F^[lFa[lF_[lFa[l-%+AXESLABELSG6$Q!6\"F]il-%*THIC
KNESSG6#Fhz-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 109 "plot3d([x,pl(f,-1,2,5)(x)*cos(t),pl(f,-1
,2,5)(x)*sin(t)], x=-1..2, t=0..2*Pi, axes=boxed, lightmodel=light1);
" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%MESHG6#X.
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{PARA 4 "" 0 "" {TEXT -1 10 "Question 2" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 159 "Experiment with different numbers of subintervals to see
 how the piecewise-linear surfaces approach the exact surface of revol
ution.  Repeat with the function " }{XPPEDIT 18 0 "g;" "6#%\"gG" }
{TEXT -1 17 " from Question 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 9 "Solution." }{TEXT -1 51 " \+
 Here are the piecewise-linear approximations for " }{XPPEDIT 18 0 "f;
" "6#%\"fG" }{TEXT -1 54 ", with 3, 6, and 10 subintervals, revolved a
round the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 "-axis:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "plot3d([x,pl(f,-1,2,3)(x)*c
os(t),pl(f,-1,2,3)(x)*sin(t)], x=-1..2, t=0..2*Pi, axes=boxed, lightmo
del=light1);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%
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