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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Moments of Inertia " }}{PARA 0 "
" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "
" 0 "" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "
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-1 51 ", of an object consisting of particles with masses " }{XPPEDIT 
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\"\"$%(~.~.~.~G&F$6#%\"NG" }{TEXT -1 21 " about an axis AB is " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[AB] = m[1]*r[1]^2+m
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gcp6&F[tFgel$FjelF*Fgel-%*LINESTYLEG6#\"\"$-Fbel6%7$7$$\"3;+++++++FFcf
lFbep7$$\"3M+++++++**FcflFbepFgdpFjdp-Fbel6%7$7$F[\\mF[\\m7$$\"3G*****
********>)FcflF[\\mFgdpFjdp-%%TEXTG6&7$$\"#cF*$\"#YF*Q\"r6\"Fgdp-%%FON
TG6$%*HELVETICAGFcbm-F_fp6&7$$\"#pF*$\"#CF*FffpFgdpFhfp-F_fp6&7$$\"\"%
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gpF\\dp-F_fp6&7$$\"$0(F_hp$\"#BF*Q\"2FgfpFgdpFchp-F_fp6&7$$\"$:%F_hp$
\"#6F*Q\"3FgfpFgdpFchp-F_fp6&7$$FjcpFgel$\"#WF*Q\"mFgfpF[flFhfp-F_fp6&
7$$\"$1\"F*$F]epFgelF[jpF[flFhfp-F_fp6&7$$\"#lF*FadlF[jpF[flFhfp-F_fp6
&7$$FbfnF*$FgenF*Q\"AFgfpF[flFhfp-F_fp6&7$FdjnF]]lQ\"BFgfpF[flFhfp-F_f
p6&7$$\"$D*F_hpFhnFbhpF[flFchp-F_fp6&7$$\"%&3\"F_hp$\"#HF*F\\ipF[flFch
p-F_fp6&7$$\"$v'F_hp$\"#8F*FdipF[flFchp-%+AXESLABELSG6%Q!FgfpFg\\q-Fif
p6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$Fj\\qFj\\q" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" 
"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Cur
ve 23" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "If  " }{XPPEDIT 
18 0 "M = m[1]+m[2]+m[3]+` . . . `+m[N];" "6#/%\"MG,,&%\"mG6#\"\"\"F)&
F'6#\"\"#F)&F'6#\"\"$F)%(~.~.~.~GF)&F'6#%\"NGF)" }{TEXT -1 43 " is the
 total mass of the object, then the " }{TEXT 259 18 "radius of gyratio
n" }{TEXT -1 59 " of the object with respect to the axis AB is the dis
tance " }{XPPEDIT 18 0 "k[AB]" "6#&%\"kG6#%#ABG" }{TEXT -1 44 " from t
he axis at which a single point mass " }{TEXT 454 1 "M" }{TEXT -1 148 
" placed at this distance from the axis has the same moment of inertia
 (about AB) as the object consisting of the collection of particles, t
hat is,  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M*k^2=I[
AB]" "6#/*&%\"MG\"\"\"*$%\"kG\"\"#F&&%\"IG6#%#ABG" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "k^2=I[AB]/M" "6#/*$%\"kG\"\"#*&&%\"IG6#%#ABG\"\"\"%
\"MG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 259 4 "Note" }{TEXT -1 70 ": Suppose that the object is \+
rotating about the axis AB with constant " }{TEXT 259 16 "angular velo
city" }{TEXT -1 1 " " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 
80 ". Then the actual velocities of the individual particles with resp
ective masses " }{XPPEDIT 18 0 "m[1],m[2],m[3],` . . . `,m[N];" "6'&%
\"mG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F$6#%\"NG" }{TEXT -1 5 " ar
e " }{XPPEDIT 18 0 "v[1] = r[1]*omega,v[2] = r[2]*omega,` . . . `,v[N]
 = r[N]*omega;" "6&/&%\"vG6#\"\"\"*&&%\"rG6#F'F'%&omegaGF'/&F%6#\"\"#*
&&F*6#F0F'F,F'%(~.~.~.~G/&F%6#%\"NG*&&F*6#F8F'F,F'" }{TEXT -1 6 ". The
 " }{TEXT 259 14 "kinetic energy" }{TEXT -1 18 " of the object is " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "E=1/2" "6#/%\"EG*&\"
\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m[1]*v[1]^2+1/2" "6#
,&*&&%\"mG6#\"\"\"F(*$&%\"vG6#F(\"\"#F(F(*&F(F(F-!\"\"F(" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "m[1]*v[1]^2+` . . . `+1/2" "6#,(*&&%\"mG6#\"\"\"F(
*$&%\"vG6#F(\"\"#F(F(%(~.~.~.~GF(*&F(F(F-!\"\"F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "m[N]*v[N]^2;" "6#*&&%\"mG6#%\"NG\"\"\"*$&%\"vG6#F'\"\"#
F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
m[1]*r[1]^2*omega^2+1/2" "6#,&*(&%\"mG6#\"\"\"F(*$&%\"rG6#F(\"\"#F(%&o
megaGF-F(*&F(F(F-!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m[2]*r[2]^2*
omega^2+` . . . `+1/2" "6#,(*(&%\"mG6#\"\"#\"\"\"*$&%\"rG6#F(F(F)%&ome
gaGF(F)%(~.~.~.~GF)*&F)F)F(!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m[
N]*v[N]^2*omega^2;" "6#*(&%\"mG6#%\"NG\"\"\"*$&%\"vG6#F'\"\"#F(%&omega
GF-" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "(m[1]*r[1]^2+m[2]*r[2]^2+m[3]*r[3]^2+` . . . `+m[N]*r[N]^2)*omega^2
" "6#*&,,*&&%\"mG6#\"\"\"F)*$&%\"rG6#F)\"\"#F)F)*&&F'6#F.F)*$&F,6#F.F.
F)F)*&&F'6#\"\"$F)*$&F,6#F8F.F)F)%(~.~.~.~GF)*&&F'6#%\"NGF)*$&F,6#F@F.
F)F)F)*$%&omegaGF.F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "I[AB]*omega^2" "6#*&&%\"IG6#%#ABG\"\"\"*$%&omegaG\"
\"#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Moment of i
nertia of a uniform rod " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 62 "We determine the moment of inertia o
f a uniform rod of length " }{TEXT 288 1 "a" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 79 "     (i) about an axis perpendicular to the rod
 through the centre of the rod, " }}{PARA 0 "" 0 "" {TEXT -1 76 "     \+
(ii) about an axis at one end of the rod and perpendicular to the rod.
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Supp
ose that the mass per unit length, or " }{TEXT 259 14 "linear density
" }{TEXT -1 16 ", of the rod is " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "(i) Consider the rod to
 lie along the " }{TEXT 286 1 "x" }{TEXT -1 94 " axis with its centre \+
at the origin and subdivide the rod into small sections of equal lengt
h " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 285 1 "x" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 56 "We calculate the moment of inertia \+
of the rod about the " }{TEXT 287 1 "y" }{TEXT -1 7 " axis. " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 514 185 185 {PLOTDATA 2 "66-%'CU
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$6&7$7$F=$!\"(F)7$F=$\"\"(F)7%7$$!+++++D!#6$\"+++++j!#5FH7$$\"+++++DFO
FP-%&STYLEG6#%,PATCHNOGRIDGF5-F$6&7$7$F2F=7$$\"#:F)F=7%7$$\"++++]9!\"*
$\"+++++NFOFhn7$F]o$!+++++NFOFVF5-F$6&7$7$$!\"#F)$\"#NFjo7$$\"#5F)F[p7
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+7$FirF/F5F]r-F$6%7$7$$\"3;+++++++_F\\rF+7$F`sF/F5F]r-F$6%7$7$$\"3!***
************[F\\rF+7$Fgs$!3/+++++++5F\\rF5F]r-%)POLYGONSG6%7&7$$\"#YFj
o$!\"&Fjo7$Fat$\"\"&Fjo7$$\"#_FjoFft7$FitFct-%&COLORG6&F8FIFI$\"\"*F)F
V-%%TEXTG6&7$$\"#\\Fjo$!#:FjoQ\"x6\"F5-%%FONTG6$%*HELVETICAGF_p-Fbu6&7
$$FGFjo$\"#lFjoQ\"yFjuF5F[v-Fbu6&7$$!\"$F)$\"#OFjoQ\"aFjuF5F[v-Fbu6&7$
$\"#^Fjo$F`rF)FiuF5F[v-Fbu6&7$$\"#ZFjoFcwQ\"dFjuF5-F\\v6$%'SYMBOLGF_p-
%+AXESLABELSG6%Q!FjuF`x-F\\v6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG
6$FcxFcx" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur
ve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14
" "Curve 15" "Curve 16" "Curve 17" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 37 "A small section of the rod of length " }{XPPEDIT 18 0 "de
lta" "6#%&deltaG" }{TEXT 295 1 "x" }{TEXT -1 35 " has a moment of iner
tia about the " }{TEXT 296 1 "y" }{TEXT -1 9 " axis of " }{XPPEDIT 18 
0 "``(rho*delta*x)*`.`*x^2=rho*x^2*delta" "6#/*(-%!G6#*(%$rhoG\"\"\"%&
deltaGF*%\"xGF*F*%\".GF*F,\"\"#*(F)F**$F,F.F*F+F*" }{TEXT 289 1 "x" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "If the total moment of \+
inertia about the " }{TEXT 292 1 "y" }{TEXT -1 9 " axis is " }
{XPPEDIT 18 0 "I[y]" "6#&%\"IG6#%\"yG" }{TEXT -1 7 ", then " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[y]" "6#&%\"IG6#%\"yG" }
{TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(rh
o*x^2*delta*x,x=-a/2..a/2)" "6#-%$SumG6$**%$rhoG\"\"\"*$%\"xG\"\"#F(%&
deltaGF(F*F(/F*;,$*&%\"aGF(F+!\"\"F2*&F1F(F+F2" }{TEXT -1 3 ",  " }}
{PARA 0 "" 0 "" {TEXT -1 42 "where the summation is over a sequence of
 " }{TEXT 297 1 "x" }{TEXT -1 56 " values which give the locations of \+
the small sections. " }}{PARA 0 "" 0 "" {TEXT -1 91 "Passing to the li
mit where the number of small sections tends to infinity and their len
gth " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 291 1 "x" }{TEXT -1 
83 " tends to zero enables us to find the moment of inertia by an inte
gral as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
I[y]=Limit(``,delta*x=0)" "6#/&%\"IG6#%\"yG-%&LimitG6$%!G/*&%&deltaG\"
\"\"%\"xGF/\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(rho*x^2*delta*x,
x=-a/2..a/2)" "6#-%$SumG6$**%$rhoG\"\"\"*$%\"xG\"\"#F(%&deltaGF(F*F(/F
*;,$*&%\"aGF(F+!\"\"F2*&F1F(F+F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(rho*x^2,x=-a/2..a/2)" "6#/%!G-%
$IntG6$*&%$rhoG\"\"\"*$%\"xG\"\"#F*/F,;,$*&%\"aGF*F-!\"\"F3*&F2F*F-F3
" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "`` = rho*Int(x^2,x = -a/2 .. a/2);
" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$*$%\"xG\"\"#/F,;,$*&%\"aGF'F-!\"\"F3*&
F2F'F-F3F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= rho*x^3/3" "6#/%!G*(%$rhoG\"\"\"*$%\"xG\"\"$F'F*!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a/2,``],[-a/2,``])" "6#
-%*PIECEWISEG6$7$*&%\"aG\"\"\"\"\"#!\"\"%!G7$,$*&F(F)F*F+F+F," }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= rho*(a
^3/24-(-a^3/24))" "6#/%!G*&%$rhoG\"\"\",&*&%\"aG\"\"$\"#C!\"\"F',$*&F*
F+F,F-F-F-F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=rho*a^3/12" "6#/%!G*(%$rhoG\"\"\"*$%\"aG\"\"$F'\"#7!
\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "If the mass of t
he rod is " }{TEXT 293 1 "M" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "M=r
ho*a" "6#/%\"MG*&%$rhoG\"\"\"%\"aGF'" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "rho=M/a" "6#/%$rhoG*&%\"MG\"\"\"%\"aG!\"\"" }{TEXT -1 
5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[y]=M*a^
2/12" "6#/&%\"IG6#%\"yG*(%\"MG\"\"\"*$%\"aG\"\"#F*\"#7!\"\"" }{TEXT 
-1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 294 6 "______" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 47 "If the radius of gyration of the rod about the " }{TEXT 
301 1 "y" }{TEXT -1 58 " axis (which is the perpendicular bisector of \+
the rod) is " }{XPPEDIT 18 0 "k[y]" "6#&%\"kG6#%\"yG" }{TEXT -1 8 ",  \+
then " }{XPPEDIT 18 0 "k[y]^2 = a^2/12;" "6#/*$&%\"kG6#%\"yG\"\"#*&%\"
aGF)\"#7!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "k[y]=a/(2*sqrt(
3))" "6#/&%\"kG6#%\"yG*&%\"aG\"\"\"*&\"\"#F*-%%sqrtG6#\"\"$F*!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "(ii) Consider the rod to lie along the " }{TEXT 298 1 "x
" }{TEXT -1 39 " axis with its left end at the origin. " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{GLPLOT2D 489 186 186 {PLOTDATA 2 "66-%'CURVESG
6$7'7$$!\"\"\"\"!$!3G+++++++]!#>7$F($\"3G+++++++]F-7$$\"\"\"F*F/7$F2F+
F'-%'COLOURG6&%$RGBGF*F*F*-F$6$7$7$F($F*F*7$$!3++++++++:!#<F=F5-F$6&7$
7$F($!\"(F)7$F($\"\"(F)7%7$$!++++D5!\"*$\"+++++j!#5FH7$$!++++](*FRFP-%
&STYLEG6#%,PATCHNOGRIDGF5-F$6&7$7$F2F=7$$\"#:F)F=7%7$$\"++++]9FO$\"+++
++N!#6Fhn7$F]o$!+++++NFaoFVF5-F$6&7$7$$F3F)$\"#N!\"#7$$\"#5F)Fjo7%7$$
\"++++g%*FR$\"+++++QFRF]p7$Fbp$\"+++++KFRFVF5-F$6&7$7$$F)F)Fjo7$$FRF)F
jo7%7$$!++++g%*FRFgpF^q7$FbqFdpFVF5-F$6%7$F17$F2$\"3A+++++++S!#=F5-%*L
INESTYLEG6#\"\"#-F$6%7$F.7$F(FiqF5F\\r-F$6%7$7$$\"3=+++++++YF[rF+7$Fhr
F/F5F\\r-F$6%7$7$$\"3;+++++++_F[rF+7$F_sF/F5F\\r-F$6%7$7$$\"3!********
*******[F[rF+7$Ffs$!3/+++++++5F[rF5F\\r-%)POLYGONSG6%7&7$$\"#YF\\p$!\"
&F\\p7$F`t$\"\"&F\\p7$$\"#_F\\pFet7$FhtFbt-%&COLORG6&F8FIFI$\"\"*F)FV-
%%TEXTG6&7$$\"#\\F\\p$!#:F\\pQ\"x6\"F5-%%FONTG6$%*HELVETICAGF_p-Fau6&7
$$!$2\"F\\p$\"#lF\\pQ\"yFiuF5Fju-Fau6&7$F=$\"#OF\\pQ\"aFiuF5Fju-Fau6&7
$$\"#^F\\p$F_rF)FhuF5Fju-Fau6&7$$\"#ZF\\pFawQ\"dFiuF5-F[v6$%'SYMBOLGF_
p-%+AXESLABELSG6%Q!FiuF^x-F[v6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEW
G6$FaxFax" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu
rve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14
" "Curve 15" "Curve 16" "Curve 17" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 50 "Subdividing the rod into small sections of length " }
{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 299 1 "x" }{TEXT -1 18 " as \+
before gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[y
]=Limit(``,delta*x=0)" "6#/&%\"IG6#%\"yG-%&LimitG6$%!G/*&%&deltaG\"\"
\"%\"xGF/\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(rho*x^2*delta*x,x \+
= 0 .. a);" "6#-%$SumG6$**%$rhoG\"\"\"*$%\"xG\"\"#F(%&deltaGF(F*F(/F*;
\"\"!%\"aG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Int(rho*x^2,x =0 .. a)" "6#/%!G-%$IntG6$*&%$rhoG\"
\"\"*$%\"xG\"\"#F*/F,;\"\"!%\"aG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``= rho*x^3/3" "6#/%!G*(%$rhoG\"\"\"*$%
\"xG\"\"$F'F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a, ``],
[0, ``]);" "6#-%*PIECEWISEG6$7$%\"aG%!G7$\"\"!F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "``=rho*a^3/3" "6#/%!G*(%$rhoG\"\"\"*$%\"aG\"\"$F'F*!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[y] = M*a^2/3;" "6#/&%\"IG
6#%\"yG*(%\"MG\"\"\"*$%\"aG\"\"#F*\"\"$!\"\"" }{TEXT -1 2 ". " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 300 6 "______" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "If th
e radius of gyration of the rod about the " }{TEXT 302 1 "y" }{TEXT 
-1 56 " axis (which is perpendicular to the rod at one end) is " }
{XPPEDIT 18 0 "k[y]" "6#&%\"kG6#%\"yG" }{TEXT -1 8 ",  then " }
{XPPEDIT 18 0 "k[y]^2 = a^2/3;" "6#/*$&%\"kG6#%\"yG\"\"#*&%\"aGF)\"\"$
!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "k[y] = a/sqrt(3);" "6#/
&%\"kG6#%\"yG*&%\"aG\"\"\"-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "
Moment of inertia of a disc" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 94 "We determine the moment of inerti
a of a uniform lamina in the shape of a plane disc of radius " }{TEXT 
308 1 "R" }{TEXT -1 84 " about an axis perpendicular to the disc and p
assing through the centre of the disc." }}{PARA 256 "" 0 "" {TEXT -1 
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6&7#Fj\\q-%'SYMBOLG6#%'CIRCLEGF[^l-Fgio6#%&POINTG-F$6&F[ar-F]ar6#%(DIA
MONDGF[^lF`ar-F$6&F[ar-F]ar6#%&CROSSGF[^lF`ar-F$6&7$7$$\"+++++SF`amF+7
$$\"+++++qF`amF+7%7$$\"+++++nF`am$Fc^qFcamFcbr7$Fhbr$!+++++DFcamFfioF[
^l-F$6&7$7$$!#iF\\bmF+7$$!#qF\\bmF+7%7$$!++++?nF`am$!+++++?FcamFdcr7$F
icr$\"+++++?FcamFfioF[^l-F$6&7$7$$!#\")F\\bmF+7$F[enF+7%7$$!++++!e(F`a
mF^drFfdr7$FidrF[drFfioF[^l-%%TEXTG6&7$$\"+++++NF`amF+Q\"r6\"F[^l-%%FO
NTG6$%*HELVETICAG\"#5-F]er6&7$$!#bF\\bm$FeioF\\bmFberF[^lFder-F]er6&7$
$!#eF\\bmF^frQ\"dFcerF[^l-Feer6$F]arFher-%+AXESLABELSG6%Q!FcerFjfr-Fee
r6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$F]grF]gr" 1 2 0 1 10 0 2 
9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1
0" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "
Curve 17" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Let the mass
 per unit area, or " }{TEXT 259 12 "area density" }{TEXT -1 24 ", of t
he disc lamina be " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 85 "Suppose that the disc is subdivided into
 a large number of concentric rings of width " }{XPPEDIT 18 0 "delta" 
"6#%&deltaG" }{TEXT 304 1 "r" }{TEXT -1 38 ", with a typical ring havi
ng a radius " }{TEXT 305 1 "r" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 182 "If the disc is very narrow it does not matter whether th
is is the inner or outer radius of the ring, or whether the radius is \+
perhaps measured to a central circle of the narrow ring." }}{PARA 0 "
" 0 "" {TEXT -1 33 "The ring has an approximate area " }{XPPEDIT 18 0 
"2*Pi*r*delta" "6#**\"\"#\"\"\"%#PiGF%%\"rGF%%&deltaGF%" }{TEXT 306 1 
"r" }{TEXT -1 36 ", and therefore an approximate mass " }{XPPEDIT 18 
0 "2*Pi*r*delta;" "6#**\"\"#\"\"\"%#PiGF%%\"rGF%%&deltaGF%" }{XPPEDIT 
18 0 "r*`.`*rho" "6#*(%\"rG\"\"\"%\".GF%%$rhoGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 78 "Since all points in the \"ring element\" \+
can be regarded to be at a distance of " }{TEXT 303 1 "r" }{TEXT -1 
106 " from the central perpendicular axis, the moment of inertia of th
e ring about this axis is its mass times " }{XPPEDIT 18 0 "r^2" "6#*$%
\"rG\"\"#" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "2*Pi*r*delta" "6
#**\"\"#\"\"\"%#PiGF%%\"rGF%%&deltaGF%" }{XPPEDIT 18 0 "r*rho*`.`*r^2 \+
= 2*Pi*r^3*rho*delta;" "6#/**%\"rG\"\"\"%$rhoGF&%\".GF&F%\"\"#*,F)F&%#
PiGF&F%\"\"$F'F&%&deltaGF&" }{TEXT 307 1 "r" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 87 "The moment of inertia of the disc lamina about \+
the perpendicular axis is approximately " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "Sum(2*Pi*rho*r^3*delta,r = 0 .. R);" "6#-%$Su
mG6$*,\"\"#\"\"\"%#PiGF(%$rhoGF(%\"rG\"\"$%&deltaGF(/F+;\"\"!%\"RG" }
{TEXT 312 1 "r" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+
" }{TEXT 310 1 "r" }{TEXT -1 53 " ranges over a finite number of value
s between 0 and " }{TEXT 309 1 "R" }{TEXT -1 87 ", corresponding to th
e radii of the concentric rings into which the disc is subdivided." }}
{PARA 0 "" 0 "" {TEXT -1 81 "The limit of this sum, as the number of r
ings tends to infinity, and their width " }{XPPEDIT 18 0 "delta;" "6#%
&deltaG" }{TEXT 311 1 "r" }{TEXT -1 42 " tends to zero, is the definit
e integral: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(
2*Pi*rho*r^3,r=0..R)=2*Pi*rho*Int(r^3,r=0..R)" "6#/-%$IntG6$**\"\"#\"
\"\"%#PiGF)%$rhoGF)%\"rG\"\"$/F,;\"\"!%\"RG**F(F)F*F)F+F)-F%6$*$F,F-/F
,;F0F1F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=2*Pi*rho*r^4/4" "6#/%!G*,\"\"#\"\"\"%#PiGF'%$rhoGF'%
\"rG\"\"%F+!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R,``],[0,
``])" "6#-%*PIECEWISEG6$7$%\"RG%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi*rho*R^4/2" "6#/%!G**%
#PiG\"\"\"%$rhoGF'%\"RG\"\"%\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 34 "If the mass of the disc lamina is " }{TEXT 313 1 "M
" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "M = rho*Pi*R^2;" "6#/%\"MG*(%$
rhoG\"\"\"%#PiGF'%\"RG\"\"#" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 
"rho = M/(Pi*R^2);" "6#/%$rhoG*&%\"MG\"\"\"*&%#PiGF'*$%\"RG\"\"#F'!\"
\"" }{TEXT -1 84 ".  Hence its moment of inertia about the perpendicul
ar axis through its centre is:  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "I[C] = M*R^2/2;" "6#/&%\"IG6#%\"CG*(%\"MG\"\"\"*$%\"RG
\"\"#F*F-!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 314 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 75 "If the radius of gyration of the disc ab
out the axis through its centre is " }{XPPEDIT 18 0 "k[C];" "6#&%\"kG6
#%\"CG" }{TEXT -1 8 ",  then " }{XPPEDIT 18 0 "k[C]^2 = R^2/2;" "6#/*$
&%\"kG6#%\"CG\"\"#*&%\"RGF)F)!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 
18 0 "k[C] = R/sqrt(2);" "6#/&%\"kG6#%\"CG*&%\"RG\"\"\"-%%sqrtG6#\"\"#
!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Moment of i
nertia of a rectangular lamina " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 99 "We determine the moment of in
ertia of a uniform lamina in the shape of a plane rectangle of length \+
" }{TEXT 322 1 "a" }{TEXT -1 12 " and height " }{TEXT 323 1 "b" }
{TEXT -1 48 " about an axis along one of the sides of length " }{TEXT 
445 1 "b" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
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{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 31 "Let the mass per unit \+
area, or " }{TEXT 259 12 "area density" }{TEXT -1 24 ", of the disc la
mina be " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 102 "Suppose that the rectangle lies in the first quad
rant in the coordinate plane with one side of length " }{TEXT 325 1 "a
" }{TEXT -1 11 " along the " }{TEXT 324 1 "x" }{TEXT -1 29 " axis and \+
one side of length " }{TEXT 326 1 "b" }{TEXT -1 11 " along the " }
{TEXT 327 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 66 "W
e find the moment of inertia of the rectangular lamina about the " }
{TEXT 328 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 84 "S
uppose that the disc is subdivided into a large number of vertical str
ips of width " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 316 1 "x" }
{TEXT -1 31 ", with a typical strip located " }{TEXT 330 1 "x" }{TEXT 
-1 16 " units from the " }{TEXT 331 1 "y" }{TEXT -1 7 " axis.." }}
{PARA 0 "" 0 "" {TEXT -1 177 "If the strip is very narrow it does not \+
matter whether this distance is measured to the left side of the strip
, or to the right side of the strip, or to the middle of the strip." }
}{PARA 0 "" 0 "" {TEXT -1 22 "The strip has an area " }{XPPEDIT 18 0 "
b*delta;" "6#*&%\"bG\"\"\"%&deltaGF%" }{TEXT 317 1 "x" }{TEXT -1 27 ",
 and therefore a mass of  " }{XPPEDIT 18 0 "b*delta;" "6#*&%\"bG\"\"\"
%&deltaGF%" }{XPPEDIT 18 0 "x*`.`*rho;" "6#*(%\"xG\"\"\"%\".GF%%$rhoGF
%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 78 "Since all points in the vertical strip can be regarded to
 be at a distance of " }{TEXT 315 1 "x" }{TEXT -1 10 " from the " }
{TEXT 329 1 "y" }{TEXT -1 52 " axis, the moment of inertia of the stri
p about the " }{TEXT 335 1 "y" }{TEXT -1 24 " axis is its mass times \+
" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 11 ", that is, " }
{XPPEDIT 18 0 "b*delta;" "6#*&%\"bG\"\"\"%&deltaGF%" }{XPPEDIT 18 0 "x
*rho*`.`*x^2 = b*rho*x^2*delta;" "6#/**%\"xG\"\"\"%$rhoGF&%\".GF&F%\"
\"#**%\"bGF&F'F&F%F)%&deltaGF&" }{TEXT 318 1 "x" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 61 "If the moment of inertia of the rectangul
ar lamina about the " }{TEXT 333 1 "y" }{TEXT -1 9 " axis is " }
{XPPEDIT 18 0 "I[y]" "6#&%\"IG6#%\"yG" }{TEXT -1 6 " then " }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "I[y]" "6#&%\"IG6#%\"yG" }{TEXT -1 1 " " }
{TEXT 334 1 "~" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Sum(b*rho*x^2*delta,x
 = 0 .. a);" "6#-%$SumG6$**%\"bG\"\"\"%$rhoGF(%\"xG\"\"#%&deltaGF(/F*;
\"\"!%\"aG" }{TEXT 332 1 "x" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 6 "where " }{TEXT 320 1 "x" }{TEXT -1 53 " ranges over a finite num
ber of values between 0 and " }{TEXT 319 1 "a" }{TEXT -1 95 ", corresp
onding to the locations of the vertical strips into which the rectangl
e is subdivided." }}{PARA 0 "" 0 "" {TEXT -1 91 "The limit of this sum
, as the number of vertical strips tends to infinity, and their width \+
" }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 321 1 "x" }{TEXT -1 42 "
 tends to zero, is the definite integral: " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "Int(b*rho*x^2,x = 0 .. a) = b*rho*Int(x^2,x \+
= 0 .. a);" "6#/-%$IntG6$*(%\"bG\"\"\"%$rhoGF)%\"xG\"\"#/F+;\"\"!%\"aG
*(F(F)F*F)-F%6$*$F+F,/F+;F/F0F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = b*rho*x^3/3;" "6#/%!G**%\"bG\"\"\"
%$rhoGF'%\"xG\"\"$F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([
a, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$%\"aG%!G7$\"\"!F(" }{TEXT -1 1 "
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*b*a^3/3" 
"6#/%!G**%$rhoG\"\"\"%\"bGF'%\"aG\"\"$F*!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 41 "If the mass of the rectangular lamina is \+
" }{TEXT 336 1 "M" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "M = rho*a*b;
" "6#/%\"MG*(%$rhoG\"\"\"%\"aGF'%\"bGF'" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "rho = M/(a*b);" "6#/%$rhoG*&%\"MG\"\"\"*&%\"aGF'%\"bGF'
!\"\"" }{TEXT -1 41 ".  Hence its moment of inertia about the " }
{TEXT 338 1 "y" }{TEXT -1 11 " axis is:  " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "I[y] = M*a^2/3;" "6#/&%\"IG6#%\"yG*(%\"MG\"\"
\"*$%\"aG\"\"#F*\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 337 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Note that this is exact
ly the same as the moment of inertia of a rod of length " }{TEXT 339 
1 "a" }{TEXT -1 16 " about one end. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Moment of inertia of a lamina \+
about the " }{TEXT 340 1 "x" }{TEXT -1 5 " and " }{TEXT 284 1 "y" }
{TEXT -1 6 " axis " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 120 "Suppose that we have a uniform lamina de
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r a graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 
32 " between the two vertical lines " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%
\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT 
-1 45 ". Here we assume that the graph is above the " }{TEXT 266 1 "x
" }{TEXT -1 10 " axis for " }{TEXT 263 1 "x" }{TEXT -1 9 " between " }
{TEXT 264 1 "a" }{TEXT -1 5 " and " }{TEXT 265 1 "b" }{TEXT -1 36 ", o
r at least does not go below the " }{TEXT 267 1 "x" }{TEXT -1 24 " axi
s in this interval. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 
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}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Let the mass per unit a
rea, or " }{TEXT 259 12 "area density" }{TEXT -1 19 ", of the lamina b
e " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 75 "Subdivide the region into a large number of vertical st
rips of equal width " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 268 
1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "If a strip locate
d at " }{TEXT 270 1 "x" }{TEXT -1 50 " is very narrow, we can suppose \+
that its area is  " }{XPPEDIT 18 0 "f(x)*`.`*delta;" "6#*(-%\"fG6#%\"x
G\"\"\"%\".GF(%&deltaGF(" }{TEXT 269 1 "x" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 22 "Its mass is therefore " }{XPPEDIT 18 0 "delta" "6#
%&deltaG" }{XPPEDIT 18 0 "m=rho*f(x)*delta" "6#/%\"mG*(%$rhoG\"\"\"-%
\"fG6#%\"xGF'%&deltaGF'" }{TEXT 272 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 45 "The moment of inertia of the strip about the " }
{TEXT 271 1 "y" }{TEXT -1 23 " axis is approximately " }{XPPEDIT 18 0 
"delta" "6#%&deltaG" }{XPPEDIT 18 0 "m*x^2 = rho*x^2*f(x)*delta;" "6#/
*&%\"mG\"\"\"*$%\"xG\"\"#F&**%$rhoGF&*$F(F)F&-%\"fG6#F(F&%&deltaGF&" }
{TEXT 273 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "The mo
ment of inertia " }{XPPEDIT 18 0 "I[y]" "6#&%\"IG6#%\"yG" }{TEXT -1 
25 " of the lamina about the " }{TEXT 274 1 "y" }{TEXT -1 91 " axis is
 approximately equal to the sum of the moments of inertia of all the s
trips, hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[y]
" "6#&%\"IG6#%\"yG" }{TEXT -1 1 " " }{TEXT 275 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(rho*x^2*f(x)*delta,x = a .. b);" "6#-%$SumG6$**%$rh
oG\"\"\"*$%\"xG\"\"#F(-%\"fG6#F*F(%&deltaGF(/F*;%\"aG%\"bG" }{TEXT 
276 1 "x" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 277 1 "x" }{TEXT -1 47 " ranges over a finite number of values b
etween " }{TEXT 279 1 "a" }{TEXT -1 5 " and " }{TEXT 280 1 "b" }{TEXT 
-1 92 ", corresponding to the locations of the vertical strips into wh
ich the region is subdivided." }}{PARA 0 "" 0 "" {TEXT -1 82 "The limi
t of this sum, as the number of strips tends to infinity, and their wi
dth " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 278 1 "x" }{TEXT -1 
42 " tends to zero, is the definite integral: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "I[y]=rho*Int(x^2*f(x),x=a..b)" "6#/&%\"
IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"#-%\"fG6#F/F*/F/;%\"aG%\"bG
F*" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 283 11 "
___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 129 "Since the vertical strip is narrow we may rega
rd it to be a rectangle for the purpose of finding its monent of inert
ia about the " }{TEXT 341 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "
" {TEXT -1 51 "Hence the moment of inertia of the strip about the " }
{TEXT 342 1 "x" }{TEXT -1 23 " axis is approximately " }{XPPEDIT 18 0 
"1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "rho*f(x
)^3*delta" "6#*(%$rhoG\"\"\"*$-%\"fG6#%\"xG\"\"$F%%&deltaGF%" }{TEXT 
343 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "The moment of
 inertia " }{XPPEDIT 18 0 "I[x];" "6#&%\"IG6#%\"xG" }{TEXT -1 25 " of \+
the lamina about the " }{TEXT 344 1 "x" }{TEXT -1 91 " axis is approxi
mately equal to the sum of the moments of inertia of all the strips, h
ence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[x];" "6#&%
\"IG6#%\"xG" }{TEXT -1 1 " " }{TEXT 345 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(1/3,x = a .. b);" "6#-%$SumG6$*&\"\"\"F'\"\"$!\"\"/
%\"xG;%\"aG%\"bG" }{TEXT 346 1 " " }{XPPEDIT 18 0 "rho*f(x)^3*delta;" 
"6#*(%$rhoG\"\"\"*$-%\"fG6#%\"xG\"\"$F%%&deltaGF%" }{TEXT 353 1 "x" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 347 1 "x
" }{TEXT -1 47 " ranges over a finite number of values between " }
{TEXT 349 1 "a" }{TEXT -1 5 " and " }{TEXT 350 1 "b" }{TEXT -1 92 ", c
orresponding to the locations of the vertical strips into which the re
gion is subdivided." }}{PARA 0 "" 0 "" {TEXT -1 82 "The limit of this \+
sum, as the number of strips tends to infinity, and their width " }
{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 348 1 "x" }{TEXT -1 42 " te
nds to zero, is the definite integral: " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "I[x] = rho*Int(f(x)^3/3,x = a .. b);" "6#/&%\"IG
6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&-%\"fG6#F'\"\"$F2!\"\"/F';%\"aG%\"bGF*
" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 351 11 "__
_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 15 "The total mass " }{TEXT 281 1 "M" }{TEXT -1 33 " o
f the lamina is its area times " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }
{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
M=rho*Int(f(x),x=a..b)" "6#/%\"MG*&%$rhoG\"\"\"-%$IntG6$-%\"fG6#%\"xG/
F.;%\"aG%\"bGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This \+
gives " }{XPPEDIT 18 0 "rho=M/Int(f(x),x = a .. b)" "6#/%$rhoG*&%\"MG
\"\"\"-%$IntG6$-%\"fG6#%\"xG/F.;%\"aG%\"bG!\"\"" }{TEXT -1 10 " so tha
t  " }{XPPEDIT 18 0 "I[x] = M*Int(f(x)^3/3,x = a .. b)/Int(f(x),x = a \+
.. b)" "6#/&%\"IG6#%\"xG*(%\"MG\"\"\"-%$IntG6$*&-%\"fG6#F'\"\"$F2!\"\"
/F';%\"aG%\"bGF*-F,6$-F06#F'/F';F6F7F3" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "I[y]=M*Int(x^2*f(x),x = a .. b) / Int(f(x),x = a .. b)
" "6#/&%\"IG6#%\"yG*(%\"MG\"\"\"-%$IntG6$*&%\"xG\"\"#-%\"fG6#F/F*/F/;%
\"aG%\"bGF*-F,6$-F26#F//F/;F6F7!\"\"" }{TEXT -1 4 ".   " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "If the radii of gyra
tion of the lamina about the " }{TEXT 352 1 "x" }{TEXT -1 16 " and y a
xes are " }{XPPEDIT 18 0 "k[x]" "6#&%\"kG6#%\"xG" }{TEXT -1 5 " and " 
}{XPPEDIT 18 0 "k[y]" "6#&%\"kG6#%\"yG" }{TEXT -1 20 " respectively th
en: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k[x]^2 = Int(
f(x)^3/3,x = a .. b)/Int(f(x),x = a .. b);" "6#/*$&%\"kG6#%\"xG\"\"#*&
-%$IntG6$*&-%\"fG6#F(\"\"$F2!\"\"/F(;%\"aG%\"bG\"\"\"-F,6$-F06#F(/F(;F
6F7F3" }{TEXT -1 9 "   and   " }{XPPEDIT 18 0 "k[y]^2=Int(x^2*f(x),x =
 a .. b)/Int(f(x),x = a .. b)" "6#/*$&%\"kG6#%\"yG\"\"#*&-%$IntG6$*&%
\"xGF)-%\"fG6#F/\"\"\"/F/;%\"aG%\"bGF3-F,6$-F16#F//F/;F6F7!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 131 "Note that the radius of gyration is does not depend on t
he density and is a property which depends on the shape of the plane a
lone." }}{PARA 0 "" 0 "" {TEXT -1 14 "The integrals " }{XPPEDIT 18 0 "
Int(f(x)^3/3,x = a .. b);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"$F+!\"\"/F*;
%\"aG%\"bG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "Int(x^2*f(x),x = a .
. b)" "6#-%$IntG6$*&%\"xG\"\"#-%\"fG6#F'\"\"\"/F';%\"aG%\"bG" }{TEXT 
-1 10 " give the " }{TEXT 259 21 "second moment of area" }{TEXT -1 31 
" of the plane region about the " }{TEXT 354 1 "x" }{TEXT -1 5 " and \+
" }{TEXT 282 1 "y" }{TEXT -1 20 " axes respectively. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 367 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 371 1 "R" }
{TEXT -1 61 " the plane region in the first quadrant bounded by the li
nes " }{XPPEDIT 18 0 "y = x+1;" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 9 " and the " }
{TEXT 369 1 "x" }{TEXT -1 5 " and " }{TEXT 370 1 "y" }{TEXT -1 6 " axe
s." }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the moment of inertia of a uni
form lamima in the shape of the region " }{TEXT 374 1 "R" }{TEXT -1 
15 " (a) about the " }{TEXT 373 1 "y" }{TEXT -1 20 " axis (b) about th
e " }{TEXT 372 1 "x" }{TEXT -1 28 " axis, in terms of the mass " }
{TEXT 412 1 "M" }{TEXT -1 17 " of the lamina.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 368 8 "Solution" }{TEXT -1 2 ": \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 215 "f := x -> x+1:\np1 := plot(f(x),x=0..4,y=0..5,color=red,thick
ness=2):\np2 := plot(f(x),x=0..3,color=COLOR(RGB,.9,.9,.95),filled=tru
e):\np3 := plot([[3,0],[3,4]],color=black):\nplots[display]([p1,p2,p3]
,tickmarks=[4,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 204 204 204 
{PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"\"!F)$\"\"\"F)7$$\"3Hmmmm;')=()!#>$
\"3mmmm;')=(3\"!#<7$$\"3RLLLe'40j\"!#=$\"3WLL$e'40j6F27$$\"3mmmm;6m$[#
F6$\"3ommm6hO[7F27$$\"3fmmm;yYULF6$\"3xmmm\"yYUL\"F27$$\"3%HLL$eF>(>%F
6$\"3CLL$eF>(>9F27$$\"3Qmmm\">K'*)\\F6$\"3kmm;>K'*)\\\"F27$$\"3P*****
\\Kd,\"eF6$\"3/++]Kd,\"e\"F27$$\"3-mmm\"fX(emF6$\"3gmm;fX(em\"F27$$\"3
.*****\\U7Y](F6$\"3!*****\\U7Y]<F27$$\"3'QLLLV!pu$)F6$\"3QLLLV!pu$=F27
$$\"3xmmm;c0T\"*F6$\"3ommmhb59>F27$$\"3#*******H,Q+5F2$\"3#*******H,Q+
?F27$$\"3)*******\\*3q3\"F2$\"3)*******\\*3q3#F27$$\"3)*******p=\\q6F2
$\"3?+++q=\\q@F27$$\"3mmm;fBIY7F2$\"3mmm;fBIYAF27$$\"3GLLLj$[kL\"F2$\"
30LLLj$[kL#F27$$\"3?LLL`Q\"GT\"F2$\"3?LLL`Q\"GT#F27$$\"3!*****\\s]k,:F
2$\"3o****\\s]k,DF27$$\"39LLL`dF!e\"F2$\"3#HLLLvv-e#F27$$\"33++]sgam;F
2$\"33++]sgamEF27$$\"3/++]<ep[<F2$\"3!)****\\<ep[FF27$$\"3QLLLe/TM=F2$
\"3;LLLe/TMGF27$$\"3JLL$eDBJ\">F2$\"3JLL$eDBJ\"HF27$$\"3immmTc-)*>F2$
\"3immmTc-)*HF27$$\"3Mmm;f`@'3#F2$\"3Mmm;f`@'3$F27$$\"3y****\\nZ)H;#F2
$\"3y****\\nZ)H;$F27$$\"3YmmmJy*eC#F2$\"3YmmmJy*eC$F27$$\"3')******R^b
JBF2$\"3')******R^bJLF27$$\"3f*****\\5a`T#F2$\"3f*****\\5a`T$F27$$\"3o
****\\7RV'\\#F2$\"3o****\\7RV'\\$F27$$\"3k*****\\@fke#F2$\"3k*****\\@f
ke$F27$$\"3/LLL`4NnEF2$\"3/LLL`4NnOF27$$\"3#*******\\,s`FF2$\"3#******
*\\,s`PF27$$\"3[mm;zM)>$GF2$\"3[mm;zM)>$QF27$$\"3$*******pfa<HF2$\"3$*
******pfa<RF27$$\"3#HLLeg`!)*HF2$\"3#HLLeg`!)*RF27$$\"3w****\\#G2A3$F2
$\"3w****\\#G2A3%F27$$\"3;LLL$)G[kJF2$\"3;LLL$)G[kTF27$$\"3#)****\\7yh
]KF2$\"3#)****\\7yh]UF27$$\"3xmmm')fdLLF2$\"3Kmmm')fdLVF27$$\"3bmmm,FT
=MF2$\"36mmm,FT=WF27$$\"3FLL$e#pa-NF2$\"3FLL$e#pa-XF27$$\"3!*******Rv&
)zNF2$\"3!*******Rv&)zXF27$$\"3ILLLGUYoOF2$\"3%GLL$GUYoYF27$$\"3_mmm1^
rZPF2$\"33mmm1^rZZF27$$\"34++]sI@KQF2$\"34++]sI@K[F27$$\"34++]2%)38RF2
$\"34++]2%)38\\F27$$\"\"%F)$\"\"&F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F
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Fg\\l7$$\"3A++]i3&o]#F6$\"33++D'3&o]7F27$F_]lF)7&Fc]lF^]l7$$\"3%)***\\
(oX*y9$F6$\"3/+](oX*y98F27$Ff]lF)7&Fj]lFe]l7$$\"3z***\\P9CAu$F6$\"3#**
*\\P9CAu8F27$F]^lF)7&Fa^lF\\^l7$$\"3!)***\\P*zhdVF6$\"3.+]P*zhdV\"F27$
Fd^lF)7&Fh^lFc^l7$$\"31++v$>fS*\\F6$\"3++]P>fS*\\\"F27$F[_lF)7&F__lFj^
l7$$\"3$)***\\(=$f%GcF6$\"35+](=$f%Gc\"F27$Fb_lF)7&Ff_lFa_l7$$\"3Q+++D
y,\"G'F6$\"3$*****\\#y,\"G;F27$Fi_lF)7&F]`lFh_l7$$\"33++]7<zboF6$\"37+
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Fg`lF)7&F[alFf`l7$$\"3!)*****\\7nD:)F6$\"34++]7nD:=F27$F^alF)7&FbalF]a
l7$$\"3[+++D!*oy()F6$\"3%*****\\-*oy(=F27$FealF)7&FialFdal7$$\"3))***
\\PpnsM*F6$\"35+]PpnsM>F27$F\\blF)7&F`blF[bl7$$\"3,++]siL-5F2$\"3,++]s
iL-?F27$FcblF)7&FgblFbbl7$$\"3-+++!R5'f5F2$\"3-+++!R5'f?F27$FjblF)7&F^
clFibl7$$\"3)***\\P/QBE6F2$\"3)***\\P/QBE@F27$FaclF)7&FeclF`cl7$$\"3!*
*****\\\"o?&=\"F2$\"3!******\\\"o?&=#F27$FhclF)7&F\\dlFgcl7$$\"31+]Pa&
4*\\7F2$\"3%)**\\Pa&4*\\AF27$F_dlF)7&FcdlF^dl7$$\"33+]7j=_68F2$\"3&)**
\\7j=_6BF27$FfdlF)7&FjdlFedl7$$\"33++vVy!eP\"F2$\"33++vVy!eP#F27$F]elF
)7&FaelF\\el7$$\"34+](=WU[V\"F2$\"3K+](=WU[V#F27$FdelF)7&FhelFcel7$$\"
3)****\\7B>&)\\\"F2$\"3)****\\7B>&)\\#F27$F[flF)7&F_flFjel7$$\"3)***\\
P>:mk:F2$\"3)***\\P>:mkDF27$FbflF)7&FfflFafl7$$\"3'***\\iv&QAi\"F2$\"3
'***\\iv&QAi#F27$FiflF)7&F]glFhfl7$$\"31++vtLU%o\"F2$\"31++vtLU%o#F27$
F`glF)7&FdglF_gl7$$\"3!******\\Nm'[<F2$\"37+++bjm[FF27$FgglF)7&F[hlFfg
l7$$\"3\"****\\(yb^6=F2$\"3\"****\\(yb^6GF27$F^hlF)7&FbhlF]hl7$$\"3)**
*\\PMaKs=F2$\"3)***\\PMaKsGF27$FehlF)7&FihlFdhl7$$\"3&****\\7TW)R>F2$
\"3=++D6W%)RHF27$F\\ilF)7&F`ilF[il7$$\"3z*****\\@80+#F2$\"3z*****\\@80
+$F27$FcilF)7&FgilFbil7$$\"31++]7,Hl?F2$\"31++]7,HlIF27$FjilF)7&F^jlFi
il7$$\"3()**\\P4w)R7#F2$\"3()**\\P4w)R7$F27$FajlF)7&FejlF`jl7$$\"3;++]
x%f\")=#F2$\"3;++]x%f\")=$F27$FhjlF)7&F\\[mFgjl7$$\"3!)**\\P/-a[AF2$\"
3!)**\\P/-a[KF27$F_[mF)7&Fc[mF^[m7$$\"3/+](=Yb;J#F2$\"3/+](=Yb;J$F27$F
f[mF)7&Fj[mFe[m7$$\"3')****\\i@OtBF2$\"3')****\\i@OtLF27$F]\\mF)7&Fa\\
mF\\\\m7$$\"3')**\\PfL'zV#F2$\"3')**\\PfL'zV$F27$Fd\\mF)7&Fh\\mFc\\m7$
$\"3>+++!*>=+DF2$\"3>+++!*>=+NF27$F[]mF)7&F_]mFj\\m7$$\"3-++DE&4Qc#F2$
\"3-++DE&4Qc$F27$Fb]mF)7&Ff]mFa]m7$$\"3=+]P%>5pi#F2$\"3=+]P%>5pi$F27$F
i]mF)7&F]^mFh]m7$$\"39+++bJ*[o#F2$\"39+++bJ*[o$F27$F`^mF)7&Fd^mF_^m7$$
\"33++Dr\"[8v#F2$\"33++Dr\"[8v$F27$Fg^mF)7&F[_mFf^m7$$\"3++++Ijy5GF2$
\"3++++Ijy5QF27$F^_mF)7&Fb_mF]_m7$$\"31+]P/)fT(GF2$\"31+]P/)fT(QF27$Fe
_mF)7&Fi_mFd_m7$$\"31+]i0j\"[$HF2$\"31+]i0j\"[$RF27$F\\`mF)7&F``mF[`m7
$$\"\"$F)Fez7$Fc`mF)7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&F\\[l$\"\"*
!\"\"F^am$\"#&*!\"#-F$6$7$7$Fc`mF(Fb`m-Fjz6&F\\[lF)F)F)-%*AXESTICKSG6$
FfzFhz-%+AXESLABELSG6%Q\"x6\"Q\"yFabm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(
Fez;F(Fgz" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "L
et " }{XPPEDIT 18 0 "f(x) = x+1;" "6#/-%\"fG6#%\"xG,&F'\"\"\"F)F)" }
{TEXT -1 45 " and let the (area) density of the lamina be " }{XPPEDIT 
18 0 "rho" "6#%$rhoG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "T
he area of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 3);" "6#/%\"AG-%$IntG6$-%\"fG6#%
\"xG/F+;\"\"!\"\"$" }{XPPEDIT 18 0 "`` = Int(x+1,x = 0 .. 3);" "6#/%!G
-%$IntG6$,&%\"xG\"\"\"F*F*/F);\"\"!\"\"$" }{TEXT -1 1 " " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x^2/2+x;" "6#/%!G,&*&%\"xG
\"\"#F(!\"\"\"\"\"F'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, \+
``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"!F(" }{XPPEDIT 18 0 "
`` = 9/2+3;" "6#/%!G,&*&\"\"*\"\"\"\"\"#!\"\"F(\"\"$F(" }{XPPEDIT 18 
0 "`` = 15/2;" "6#/%!G*&\"#:\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 26 "The mass of the lamina is " }{XPPEDIT 18 
0 "M = 15/2*rho;" "6#/%\"MG*(\"#:\"\"\"\"\"#!\"\"%$rhoGF'" }{TEXT -1 
10 ", so that " }{XPPEDIT 18 0 "rho = 2*M/15;" "6#/%$rhoG*(\"\"#\"\"\"
%\"MGF'\"#:!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 50 "(a) The moment of inertia of the region a
bout the " }{TEXT 375 1 "y" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "I[y] = rho*Int(x^2*f(x),x = 0 .. \+
3);" "6#/&%\"IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"#-%\"fG6#F/F*/
F/;\"\"!\"\"$F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = rho*Int(x^2*(x+1),x = 0 .. 3);" "6#/%!G*&%$rhoG\"
\"\"-%$IntG6$*&%\"xG\"\"#,&F,F'F'F'F'/F,;\"\"!\"\"$F'" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho*Int(x^3+
x^2,x = 0 .. 3);" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$,&*$%\"xG\"\"$F'*$F-\"
\"#F'/F-;\"\"!F.F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = rho*(x^4/4+x^3/3);" "6#/%!G*&%$rhoG\"\"\",&*&%
\"xG\"\"%F+!\"\"F'*&F*\"\"$F.F,F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P
IECEWISE([3, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"!F(" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho*(81/4+9);" "6#/%!G*&%$rhoG\"\"
\",&*&\"#\")F'\"\"%!\"\"F'\"\"*F'F'" }{XPPEDIT 18 0 "`` = 117/4*rho;" 
"6#/%!G*(\"$<\"\"\"\"\"\"%!\"\"%$rhoGF'" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = ``(117/4)*``(2/15)*M;" "6#/%!G*(-F$6#*&\"$<\"\"\"\"\"\"%!\"\"F
*-F$6#*&\"\"#F*\"#:F,F*%\"MGF*" }{XPPEDIT 18 0 "`` = 39*M/10;" "6#/%!G
*(\"#R\"\"\"%\"MGF'\"#5!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(a) The moment of inertia
 of the region about the " }{TEXT 376 1 "x" }{TEXT -1 10 " axis is: " 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "I[x] = rho*Int(f(x
)^3/3,x = 0 .. 3);" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&-%\"fG6#
F'\"\"$F2!\"\"/F';\"\"!F2F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x+1)^3,x = 0 .. 3);" "6#-%$In
tG6$*$,&%\"xG\"\"\"F)F)\"\"$/F(;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$rhoG\"\"\"
\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((x+1)^4/4);" "6#-%!G6#*
&,&%\"xG\"\"\"F)F)\"\"%F*!\"\"" }{XPPEDIT 18 0 "PIECEWISE([3, ``],[0, \+
``]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"!F(" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho/3;" "6#/%!G*&%$rhoG\"\"\"\"\"$
!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(64-1/4) =rho/3" "6#/-%!G6#,&
\"#k\"\"\"*&F)F)\"\"%!\"\"F,*&%$rhoGF)\"\"$F," }{XPPEDIT 18 0 " ``(255
/4)=85*rho/4" "6#/-%!G6#*&\"$b#\"\"\"\"\"%!\"\"*(\"#&)F)%$rhoGF)F*F+" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "``=``(85/4)*``(2/15)*M" "6#/%!G*(-F$6#*&\"#&)\"\"\"\"\"
%!\"\"F*-F$6#*&\"\"#F*\"#:F,F*%\"MGF*" }{XPPEDIT 18 0 "``=17*M/6" "6#/
%!G*(\"#<\"\"\"%\"MGF'\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "f := x -
> x+1;\na := 0: b := 3:\nInt(f(x),x=a..b);\nA := value(%);\nrho*Int(x^
2*f(x),x=a..b);\nIy := value(%):\nIy = Iy*M/(A*rho);\nrho/3*Int(f(x)^3
,x=a..b);\nIx := value(%):\nIx = Ix*M/(A*rho); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"F.F.
F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&%\"xG\"\"\"F(F(/F
';\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"#:\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%$rhoG\"\"\"-%$IntG6$*&)%\"xG\"\"#F%
,&F+F%F%F%F%/F+;\"\"!\"\"$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(
\"$<\"\"\"\"\"\"%!\"\"%$rhoGF'F',$*(\"#RF'\"#5F)%\"MGF'F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%$rhoGF&-%$IntG6$*$),&%\"xG
F&F&F&F'F&/F0;\"\"!F'F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(\"
#&)\"\"\"\"\"%!\"\"%$rhoGF'F',$*(\"#<F'\"\"'F)%\"MGF'F'" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" 
{TEXT 377 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "
Let " }{TEXT 381 1 "R" }{TEXT -1 61 " the plane region in the first qu
adrant bounded by the curve " }{XPPEDIT 18 0 "y=4-x^2" "6#/%\"yG,&\"\"
%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 9 " and the " }{TEXT 379 1 "x" }
{TEXT -1 5 " and " }{TEXT 380 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 
0 "" {TEXT -1 74 "Find the moment of inertia of a uniform lamima in th
e shape of the region " }{TEXT 384 1 "R" }{TEXT -1 15 " (a) about the \+
" }{TEXT 383 1 "y" }{TEXT -1 20 " axis (b) about the " }{TEXT 382 1 "x
" }{TEXT -1 28 " axis, in terms of the mass " }{TEXT 413 1 "M" }{TEXT 
-1 17 " of the lamina.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 378 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "f := x -> 4
-x^2:\np1 := plot(f(x),x=0..2.1,y=0..4.1,color=red,thickness=2):\np2 :
= plot(f(x),x=0..2,color=COLOR(RGB,.9,.9,.95),filled=true):\nplots[dis
play]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 193 209 209 {PLOTDATA 2 
"6&-%'CURVESG6%7S7$$\"\"!F)$\"\"%F)7$$\"3O+++vBSxX!#>$\"3SM(\\(QZ!z*R!
#<7$$\"3e++D1d<g&)F/$\"3s7y=RBn#*RF27$$\"31++D'3ARI\"!#=$\"3k%*)H>(y*H
)RF27$$\"3Q++v.cza<F;$\"3s'f!*Q#p?pRF27$$\"36+]7)>EN?#F;$\"3un<%HsW9&R
F27$$\"3C+]i+pb>EF;$\"3WzQW;#z8$RF27$$\"3#***\\i&fK.0$F;$\"3Mw1c5Z&p!R
F27$$\"3g+]iN9%e\\$F;$\"3]t'pl#4zxQF27$$\"35+]7B:#*RRF;$\"3ljh\"R=qZ%Q
F27$$\"3[++]xCr'R%F;$\"3.e>!R>*o1QF27$$\"3s++v)>a!*z%F;$\"3++Y(zy!ppPF
27$$\"3O,+]#o&*>D&F;$\"3^8+^8a;CPF27$$\"3?++]()pz1dF;$\"354YV\"oCVn$F2
7$$\"3M,+]<B3XhF;$\"39)[6JjzBi$F27$$\"3[+]i&Q(3VlF;$\"3u\\2ku+)=d$F27$
$\"3K++]2RN;qF;$\"3p%4Z%yxq2NF27$$\"3w+++IFF<uF;$\"3A$)z[_1%)\\MF27$$
\"3_+]iImj$)yF;$\"3!pniZt#[yLF27$$\"3s+++0xW'H)F;$\"3+-?ya&*o6LF27$$\"
3M,]i!)oO\\()F;$\"3>BA)=z&[MKF27$$\"3)4+v=/`1=*F;$\"358rC(4cr:$F27$$\"
3J,+D1\\lI'*F;$\"3iFswg[]sIF27$$\"3F+DJ4(*Q/5F2$\"3)4:F=J,7*HF27$$\"37
+](=Yj*[5F2$\"3K:7lbcn**GF27$$\"3-+DcjIE&4\"F2$\"3)po&f@))R+GF27$$\"3-
+v$H+nb8\"F2$\"3vV&R=e([5FF27$$\"33+]ihj4z6F2$\"3(po-+xJ(4EF27$$\"38++
][k1C7F2$\"3g'fc'H8m,DF27$$\"3-+]7041o7F2$\"3mlN*3a@?R#F27$$\"37+D1/yi
58F2$\"3o3$=#fZD#G#F27$$\"37+](y3\"*yN\"F2$\"3Y['pMzJh:#F27$$\"3******
\\]#f.S\"F2$\"3(z2R&pR**Q?F27$$\"3E++vyIqX9F2$\"3RqF43E%*4>F27$$\"39+D
cE8z'[\"F2$\"3=@&e7b^%*y\"F27$$\"3M++DM;rJ:F2$\"3'H>.&p%fQl\"F27$$\"3%
**\\iI9yRd\"F2$\"3^C:;0GfA:F27$$\"3;+DJB)e\"=;F2$\"3)[!f`A?c\"Q\"F27$$
\"3=++v8NNh;F2$\"3'z_]B]/*R7F27$$\"3/+Dc^Vd1<F2$\"3YK!*e#)Rg(3\"F27$$
\"3K+++$RF,v\"F2$\"3YE-r#3T0P*F;7$$\"35+]Pomm%z\"F2$\"3?uxB%\\:<z(F;7$
$\"3@+D1Or$)Q=F2$\"31>YPq)zn='F;7$$\"3G++]3_Uz=F2$\"3=1tac)3wn%F;7$$\"
3C+]()>P%f#>F2$\"3Zk/uyyS2HF;7$$\"3G+++J/bn>F2$\"3eKrr9IX(G\"F;7$$\"39
+D1j=\">,#F2$!3#)**>;tW$*yZF/7$$\"3W+v$RTrV0#F2$!3eZc.k!>W?#F;7$$\"33+
++++++@F2$!3U,++++++TF;-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNES
SG6#\"\"#-%)POLYGONSG6U7&7$F(F)F'7$$\"39LLLL3VfVF/$\"3;Q4GO&*4)*RF27$F
j[lF)7&F^\\lFi[l7$$\"3'pmm;H[D:)F/$\"3?/_j&f`L*RF27$Fa\\lF)7&Fe\\lF`\\
l7$$\"3LLLLe0$=C\"F;$\"3/&*Qko&yX)RF27$Fh\\lF)7&F\\]lFg\\l7$$\"3ILLL3R
Br;F;$\"3#*ojBs(p?(RF27$F_]lF)7&Fc]lF^]l7$$\"3Ymm;zjf)4#F;$\"3S&[tB$*e
f&RF27$Ff]lF)7&Fj]lFe]l7$$\"3=LL$e4;[\\#F;$\"3/4(zk#*ex$RF27$F]^lF)7&F
a^lF\\^l7$$\"3p****\\i'y]!HF;$\"3Fsokz^g:RF27$Fd^lF)7&Fh^lFc^l7$$\"3,L
L$ezs$HLF;$\"3E\\O'yw_\"*)QF27$F[_lF)7&F__lFj^l7$$\"3_****\\7iI_PF;$\"
3#*Qj(3)>?fQF27$Fb_lF)7&Ff_lFa_l7$$\"3#pmmm@Xt=%F;$\"3-)ek.ShY#QF27$Fi
_lF)7&F]`lFh_l7$$\"3QLLL3y_qXF;$\"3'p^Kbv-6z$F27$F``lF)7&Fd`lF_`l7$$\"
3i******\\1!>+&F;$\"3#fHv))*)4)\\PF27$Fg`lF)7&F[alFf`l7$$\"3()******\\
Z/NaF;$\"3?u\\l&)Gg/PF27$F^alF)7&FbalF]al7$$\"3'*******\\$fC&eF;$\"3ev
fc&>([dOF27$FealF)7&FialFdal7$$\"3ELL$ez6:B'F;$\"37+RQ2Eo6OF27$F\\blF)
7&F`blF[bl7$$\"3Smmm;=C#o'F;$\"3g8f.VkZ`NF27$FcblF)7&FgblFbbl7$$\"3-mm
mm#pS1(F;$\"3$[NdRD*)4]$F27$FjblF)7&F^clFibl7$$\"3]****\\i`A3vF;$\"3f<
\"f!>bEOMF27$FaclF)7&FeclF`cl7$$\"3slmmm(y8!zF;$\"3_\"pce8#ovLF27$Fhcl
F)7&F\\dlFgcl7$$\"3V++]i.tK$)F;$\"3>1(eq/ccI$F27$F_dlF)7&FcdlF^dl7$$\"
39++](3zMu)F;$\"3e,XYMd^NKF27$FfdlF)7&FjdlFedl7$$\"3#pmm;H_?<*F;$\"3eA
$*enXteJF27$F]elF)7&FaelF\\el7$$\"3emm;zihl&*F;$\"3#oT(*>&)*)\\3$F27$F
delF)7&FhelFcel7$$\"39LLL3#G,***F;$\"3RE1\"QQt>+$F27$F[flF)7&F_flFjel7
$$\"3<LLezw5V5F2$\"3ZuUzoj#>\"HF27$FbflF)7&FfflFafl7$$\"3!****\\PQ#\\
\"3\"F2$\"3AV2*QAu.$GF27$FiflF)7&F]glFhfl7$$\"3BLL$e\"*[H7\"F2$\"3AT(G
Ct&)*QFF27$F`glF)7&FdglF_gl7$$\"3#*******pvxl6F2$\"3*\\*[GdE'4k#F27$Fg
glF)7&F[hlFfgl7$$\"3z****\\_qn27F2$\"3%G\"\\'o8;:a#F27$F^hlF)7&FbhlF]h
l7$$\"3%)***\\i&p@[7F2$\"3\"e)*H,Va>W#F27$FehlF)7&FihlFdhl7$$\"3#)****
\\2'HKH\"F2$\"3@'R&G#=dvK#F27$F\\ilF)7&F`ilF[il7$$\"3_mmmwanL8F2$\"3$*
Rz$Hs48A#F27$FcilF)7&FgilFbil7$$\"3'******\\2goP\"F2$\"3J%*4(QLcU5#F27
$FjilF)7&F^jlFiil7$$\"3CLLeR<*fT\"F2$\"3wl<V$Rn\\*>F27$FajlF)7&FejlF`j
l7$$\"3'******\\)Hxe9F2$\"33!>M#y8)>(=F27$FhjlF)7&F\\[mFgjl7$$\"3Ymm\"
H!o-*\\\"F2$\"3#yVPTk=Hv\"F27$F_[mF)7&Fc[mF^[m7$$\"3))***\\7k.6a\"F2$
\"3z\"*f#pc**\\i\"F27$Ff[mF)7&Fj[mFe[m7$$\"3emmmT9C#e\"F2$\"3c)es--7l
\\\"F27$F]\\mF)7&Fa\\mF\\\\m7$$\"3\"****\\i!*3`i\"F2$\"33HWEf4Pe8F27$F
d\\mF)7&Fh\\mFc\\m7$$\"3QLLL$*zym;F2$\"3(*R)z_y<=A\"F27$F[]mF)7&F_]mFj
\\m7$$\"3GLL$3N1#4<F2$\"3U+5F]Ohy5F27$Fb]mF)7&Ff]mFa]m7$$\"3kmm\"HYt7v
\"F2$\"34l'Q3e7/L*F;7$Fi]mF)7&F]^mFh]m7$$\"3%*******p(G**y\"F2$\"3WuGE
$)*\\:'zF;7$F`^mF)7&Fd^mF_^m7$$\"3lmm;9@BM=F2$\"390of8b#fN'F;7$Fg^mF)7
&F[_mFf^m7$$\"3ELLL`v&Q(=F2$\"3+Th:)pyl)[F;7$F^_mF)7&Fb_mF]_m7$$\"30++
DOl5;>F2$\"3)>F+uTd`G$F;7$Fe_mF)7&Fi_mFd_m7$$\"3/++v.Uac>F2$\"3u%yAxyZ
$><F;7$F\\`mF)7&F``mF[`m7$$Fc[lF)F(7$Fc`mF)7\"-%&STYLEG6#%,PATCHNOGRID
G-%&COLORG6&F\\[l$\"\"*!\"\"F]am$\"#&*!\"#-%+AXESLABELSG6%Q\"x6\"Q\"yF
gam-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#@F_am;F($\"#TF_am" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=4-x^2
" "6#/-%\"fG6#%\"xG,&\"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 45 " and let
 the (area) density of the lamina be " }{XPPEDIT 18 0 "rho" "6#%$rhoG
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The area of the regio
n is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x
),x = 0 .. 2);" "6#/%\"AG-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!\"\"#" }
{XPPEDIT 18 0 " ``= Int(4-x^2,x = 0 .. 2)" "6#/%!G-%$IntG6$,&\"\"%\"\"
\"*$%\"xG\"\"#!\"\"/F,;\"\"!F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=4*x-x^3/3" "6#/%!G,&*&\"\"%\"\"\"%\"
xGF(F(*&F)\"\"$F+!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2
,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" }{XPPEDIT 18 0 "`
`= 8-8/3" "6#/%!G,&\"\")\"\"\"*&F&F'\"\"$!\"\"F*" }{XPPEDIT 18 0 "``=1
6/3" "6#/%!G*&\"#;\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 26 "The mass of the lamina is " }{XPPEDIT 18 0 "M=16*rho/3
" "6#/%\"MG*(\"#;\"\"\"%$rhoGF'\"\"$!\"\"" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "rho=3*M/16" "6#/%$rhoG*(\"\"$\"\"\"%\"MGF'\"#;!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "(a) The moment of inertia of the region about the " }
{TEXT 385 1 "y" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "I[y]=rho*Int(x^2*f(x),x=0..2)" "6#/&%\"IG6#%
\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"#-%\"fG6#F/F*/F/;\"\"!F0F*" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 rho*Int(x^2*(4-x^2),x = 0 .. 2);" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$*&%\"
xG\"\"#,&\"\"%F'*$F,F-!\"\"F'/F,;\"\"!F-F'" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*Int(4*x^2-x^4,x=0..2
)" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$,&*&\"\"%F'*$%\"xG\"\"#F'F'*$F/F-!\"
\"/F/;\"\"!F0F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = rho*(4*x^3/3-x^5/5);" "6#/%!G*&%$rhoG\"\"\",&*(\"
\"%F'*$%\"xG\"\"$F'F-!\"\"F'*&F,\"\"&F0F.F.F'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([2,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7
$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= rho*(32/3-32/5)" "6#/%!G*&%
$rhoG\"\"\",&*&\"#KF'\"\"$!\"\"F'*&F*F'\"\"&F,F,F'" }{XPPEDIT 18 0 "``
 = 64/15*rho;" "6#/%!G*(\"#k\"\"\"\"#:!\"\"%$rhoGF'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "``=``(64/15)*``(3/16)*M" "6#/%!G*(-F$6#*&\"#k\"\"\"\"#:
!\"\"F*-F$6#*&\"\"$F*\"#;F,F*%\"MGF*" }{XPPEDIT 18 0 "``=4*M/5" "6#/%!
G*(\"\"%\"\"\"%\"MGF'\"\"&!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(a) The moment of inertia
 of the region about the " }{TEXT 386 1 "x" }{TEXT -1 10 " axis is: " 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "I[x] = rho*Int(f(x
)^3/3,x = 0 .. 2);" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&-%\"fG6#
F'\"\"$F2!\"\"/F';\"\"!\"\"#F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((4-x^2)^3,x=0..2)" "6#-%$IntG6
$*$,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"\"\"$/F+;\"\"!F," }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$
rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(64-48*x^2+12
*x^4-x^6,x = 0 .. 2);" "6#-%$IntG6$,*\"#k\"\"\"*&\"#[F(*$%\"xG\"\"#F(!
\"\"*&\"#7F(*$F,\"\"%F(F(*$F,\"\"'F./F,;\"\"!F-" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 20 "( using the formula " }{XPPEDIT 18 0 "(a+
b)^3=a^3+3*a^2*b+3*a*b^2+b^3" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"$,**$F&F)F
'*(F)F'*$F&\"\"#F'F(F'F'*(F)F'F&F'F(F.F'*$F(F)F'" }{TEXT -1 3 " ) " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$
rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(64*x-16*x^3+1
2/5*x^5-x^7/7);" "6#-%!G6#,**&\"#k\"\"\"%\"xGF)F)*&\"#;F)*$F*\"\"$F)!
\"\"*(\"#7F)\"\"&F/F*F2F)*&F*\"\"(F4F/F/" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "PIECEWISE([2, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F
(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$rhoG\"\"\"\"
\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(128-128+384/5-128/7) =128
*rho/3" "6#/-%!G6#,*\"$G\"\"\"\"F(!\"\"*&\"$%QF)\"\"&F*F)*&F(F)\"\"(F*
F**(F(F)%$rhoGF)\"\"$F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(3/5-1/7) =
128*rho/3" "6#/-%!G6#,&*&\"\"$\"\"\"\"\"&!\"\"F**&F*F*\"\"(F,F,*(\"$G
\"F*%$rhoGF*F)F," }{XPPEDIT 18 0 " ``(21/35-5/35)=128*rho/3" "6#/-%!G6
#,&*&\"#@\"\"\"\"#N!\"\"F**&\"\"&F*F+F,F,*(\"$G\"F*%$rhoGF*\"\"$F," }
{TEXT -1 1 " " }{XPPEDIT 18 0 "``(16/35) = 2048*rho/105;" "6#/-%!G6#*&
\"#;\"\"\"\"#N!\"\"*(\"%[?F)%$rhoGF)\"$0\"F+" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "``=``(2048/105)*``(3/16)*M" "6#/%!G*(-F$6#*&\"%[?\"\"\"\"$0\"!\"
\"F*-F$6#*&\"\"$F*\"#;F,F*%\"MGF*" }{XPPEDIT 18 0 "``=128*M/35" "6#/%!
G*(\"$G\"\"\"\"%\"MGF'\"#N!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Alternatively, the moment
 of inertia about the " }{TEXT 387 1 "x" }{TEXT -1 61 " axis can be de
termined by using an integral with respect to " }{TEXT 388 1 "y" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The boundary curve of th
e region can be described in the form " }{XPPEDIT 18 0 "x=sqrt(4-y)" "
6#/%\"xG-%%sqrtG6#,&\"\"%\"\"\"%\"yG!\"\"" }{XPPEDIT 18 0 "``=g(y)" "6
#/%!G-%\"gG6#%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hen
ce " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[x]=rho*Int(y
^2*g(y),y=0..4)" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"-%$IntG6$*&%\"yG\"\"#-
%\"gG6#F/F*/F/;\"\"!\"\"%F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*Int(y^2*sqrt(4-y),y=0..4)" "6#/%
!G*&%$rhoG\"\"\"-%$IntG6$*&%\"yG\"\"#-%%sqrtG6#,&\"\"%F'F,!\"\"F'/F,;
\"\"!F2F'" }{TEXT -1 9 " ...     " }{XPPEDIT 18 0 "PIECEWISE([u = 4-y,
 y=4-u],[du = -dy,y^2=16-8*u+u^2 ],[-du = dy,`y =`*0*` implies  u =`*4
 ],[``, `y =`*4*` implies  u =`*0])" "6#-%*PIECEWISEG6&7$/%\"uG,&\"\"%
\"\"\"%\"yG!\"\"/F,,&F*F+F(F-7$/%#duG,$%#dyGF-/*$F,\"\"#,(\"#;F+*&\"\"
)F+F(F+F-*$F(F7F+7$/,$F2F-F4**%$y~=GF+\"\"!F+%.~implies~~u~=GF+F*F+7$%
!G**FAF+F*F+FCF+FBF+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "``=-rho*Int((16-8*u+u^2)*sqrt(u),u=4..0)" "6#/%!G,$
*&%$rhoG\"\"\"-%$IntG6$*&,(\"#;F(*&\"\")F(%\"uGF(!\"\"*$F1\"\"#F(F(-%%
sqrtG6#F1F(/F1;\"\"%\"\"!F(F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho*Int(16*u^(1/2)-8*u^(3/2)+u^(5/
2),u = 0 .. 4);" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$,(*&\"#;F')%\"uG*&F'F'
\"\"#!\"\"F'F'*&\"\")F')F/*&\"\"$F'F1F2F'F2)F/*&\"\"&F'F1F2F'/F/;\"\"!
\"\"%F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "(since reversi
ng the limits of the integral changes its sign ) " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho" "6#/%!G%$rhoG" }{TEXT -1 1 " \+
" }{TEXT 389 1 "[" }{TEXT -1 1 " " }{XPPEDIT 18 0 "32/3" "6#*&\"#K\"\"
\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)-16/5" "6#,&)%\"u
G*&\"\"$\"\"\"\"\"#!\"\"F(*&\"#;F(\"\"&F*F*" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "u^(5/2)+2/7" "6#,&)%\"uG*&\"\"&\"\"\"\"\"#!\"\"F(*&F)F(
\"\"(F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(7/2)" "6#)%\"uG*&\"\"(\"
\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 390 1 "]" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([4,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"%%!G7
$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*(256/3-512/5+256/7)" "6#
/%!G*&%$rhoG\"\"\",(*&\"$c#F'\"\"$!\"\"F'*&\"$7&F'\"\"&F,F,*&F*F'\"\"(
F,F'F'" }{XPPEDIT 18 0 " ``=256*rho*(1/3-2/5+1/7)" "6#/%!G*(\"$c#\"\"
\"%$rhoGF',(*&F'F'\"\"$!\"\"F'*&\"\"#F'\"\"&F,F,*&F'F'\"\"(F,F'F'" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=256*rho*``((35-42+15)/105)" "6#/%!G*
(\"$c#\"\"\"%$rhoGF'-F$6#*&,(\"#NF'\"#U!\"\"\"#:F'F'\"$0\"F/F'" }
{XPPEDIT 18 0 " ``=256*rho*``(8/105)" "6#/%!G*(\"$c#\"\"\"%$rhoGF'-F$6
#*&\"\")F'\"$0\"!\"\"F'" }{XPPEDIT 18 0 "``=2048*rho/105" "6#/%!G*(\"%
[?\"\"\"%$rhoGF'\"$0\"!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 
-1 13 "which gives: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "I[x] = ``(204
8/105)*``(3/16)*M;" "6#/&%\"IG6#%\"xG*(-%!G6#*&\"%[?\"\"\"\"$0\"!\"\"F
.-F*6#*&\"\"$F.\"#;F0F.%\"MGF." }{XPPEDIT 18 0 "``=128*M/35" "6#/%!G*(
\"$G\"\"\"\"%\"MGF'\"#N!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "f := x -> 4-x^2;\na := 0: b := 2:
\nInt(f(x),x=a..b);\nA := value(%);\nrho*Int(x^2*f(x),x=a..b);\nIy := \+
value(%):\nIy = Iy*M/(A*rho);\nrho/3*Int(f(x)^3,x=a..b);\nIx := value(
%):\nIx = Ix*M/(A*rho);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%
\"xG6\"6$%)operatorG%&arrowGF(,&\"\"%\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"%\"\"\"*$)%\"xG\"\"#F(!
\"\"/F+;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"#;\"\"$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%$rhoG\"\"\"-%$IntG6$*&)%\"xG\"\"#F
%,&\"\"%F%*$F*F%!\"\"F%/F+;\"\"!F,F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/,$*(\"#k\"\"\"\"#:!\"\"%$rhoGF'F',$*(\"\"%F'\"\"&F)%\"MGF'F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%$rhoGF&-%$IntG6$*
$),&\"\"%F&*$)%\"xG\"\"#F&!\"\"F'F&/F3;\"\"!F4F&F&F&" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/,$*(\"%[?\"\"\"\"$0\"!\"\"%$rhoGF'F',$*(\"$G\"F'\"#
NF)%\"MGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl
e 3" }}{PARA 0 "" 0 "" {TEXT 355 8 "Question" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 359 1 "R" }{TEXT -1 61 " the \+
plane region in the first quadrant bounded by the curve " }{XPPEDIT 
18 0 "y = 2/(x+1);" "6#/%\"yG*&\"\"#\"\"\",&%\"xGF'F'F'!\"\"" }{TEXT 
-1 12 " , the line " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 
9 " and the " }{TEXT 357 1 "x" }{TEXT -1 5 " and " }{TEXT 358 1 "y" }
{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the moment of \+
inertia of a uniform lamima in the shape of the region " }{TEXT 362 1 
"R" }{TEXT -1 15 " (a) about the " }{TEXT 361 1 "y" }{TEXT -1 20 " axi
s (b) about the " }{TEXT 360 1 "x" }{TEXT -1 28 " axis, in terms of th
e mass " }{TEXT 414 1 "M" }{TEXT -1 17 " of the lamina.  " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 356 8 "Solution" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 277 "f := x -> 2/(x+1):\np1 := plot(f(x),x=0..1.3,y=0..2.
1,color=red,thickness=2):\np2 := plot(f(x),x=0..1,color=COLOR(RGB,.9,.
9,.95),filled=true):\np3 := plot([[1,0],[1,1]],color=black):\np4 := pl
ot([[0,1],[1,1]],color=black,linestyle=3):\nplots[display]([p1,p2,p3,p
4],tickmarks=[2,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 193 209 209 
{PLOTDATA 2 "6)-%'CURVESG6%7S7$$\"\"!F)$\"\"#F)7$$\"3[mmmT+jLG!#>$\"3P
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$$\"3@***\\7y%3TiF@$\"3Z<#Hp$[WJ7F27$FdhlF)7&FhhlFchl7$$\"35****\\P![h
Y'F@$\"3ro\\(Q:8Y@\"F27$F[ilF)7&F_ilFjhl7$$\"3kKLL$Qx$omF@$\"3709/To()
*>\"F27$FbilF)7&FfilFail7$$\"3!)*****\\P+V)oF@$\"39JF#*\\B`%=\"F27$Fii
lF)7&F]jlFhil7$$\"3?mm\"zpe*zqF@$\"37IM#*=I'4<\"F27$F`jlF)7&FdjlF_jl7$
$\"3%)*****\\#\\'QH(F@$\"332s([[zk:\"F27$FgjlF)7&F[[mFfjl7$$\"3GKLe9S8
&\\(F@$\"3aI:))4]<V6F27$F^[mF)7&Fb[mF][m7$$\"3R***\\i?=bq(F@$\"3ror'zL
\"fH6F27$Fe[mF)7&Fi[mFd[m7$$\"3\"HLL$3s?6zF@$\"3b`ZD>$>m6\"F27$F\\\\mF
)7&F`\\mF[\\m7$$\"3a***\\7`Wl7)F@$\"3g-(eHDaL5\"F27$Fc\\mF)7&Fg\\mFb\\
m7$$\"3#pmmm'*RRL)F@$\"3'>R8\\*H(34\"F27$Fj\\mF)7&F^]mFi\\m7$$\"3Qmm;a
<.Y&)F@$\"374l\\3yRy5F27$Fa]mF)7&Fe]mF`]m7$$\"3=LLe9tOc()F@$\"3+'f#=gX
Im5F27$Fh]mF)7&F\\^mFg]m7$$\"3u******\\Qk\\*)F@$\"3eB8\"p!)Ga0\"F27$F_
^mF)7&Fc^mF^^m7$$\"3CLL$3dg6<*F@$\"3QgJafOBV5F27$Ff^mF)7&Fj^mFe^m7$$\"
3ImmmmxGp$*F@$\"3))Rmp*[iD.\"F27$F]_mF)7&Fa_mF\\_m7$$\"3A++D\"oK0e*F@$
\"3\"[]l9nA9-\"F27$Fd_mF)7&Fh_mFc_m7$$\"3A++v=5s#y*F@$\"3rQlvqK)4,\"F2
7$F[`mF)7&F_`mFj_m7$$\"\"\"F)Fb`m7$Fb`mF)7\"-%&STYLEG6#%,PATCHNOGRIDG-
%&COLORG6&F\\[l$\"\"*!\"\"F]am$\"#&*!\"#-F$6$7$7$Fb`mF(Fa`m-Fjz6&F\\[l
F)F)F)-F$6%7$7$F(Fb`mFa`mFgam-%*LINESTYLEG6#\"\"$-%*AXESTICKSG6$F+F`bm
-%+AXESLABELSG6%Q\"x6\"Q\"yFhbm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#8F
_am;F($\"#@F_am" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 2/(x+1);" "6#/-%\"fG6#%\"x
G*&\"\"#\"\"\",&F'F*F*F*!\"\"" }{TEXT -1 45 " and let the (area) densi
ty of the lamina be " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 27 "The area of the region is: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 1);" 
"6#/%\"AG-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!\"\"\"" }{XPPEDIT 18 0 "`` = I
nt(2/(x+1),x = 0 .. 1);" "6#/%!G-%$IntG6$*&\"\"#\"\"\",&%\"xGF*F*F*!\"
\"/F,;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2*ln(x+1);" "6#/%!G*&\"\"#\"\"\"-%#lnG6#,&%\"xGF'F
'F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``]);" "6
#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" }{XPPEDIT 18 0 "`` = 2*ln(2);" "
6#/%!G*&\"\"#\"\"\"-%#lnG6#F&F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 26 "The mass of the lamina is " }{XPPEDIT 18 0 "M = 2*ln(2)*r
ho;" "6#/%\"MG*(\"\"#\"\"\"-%#lnG6#F&F'%$rhoGF'" }{TEXT -1 10 ", so th
at " }{XPPEDIT 18 0 "rho = M/(2*ln(2));" "6#/%$rhoG*&%\"MG\"\"\"*&\"\"
#F'-%#lnG6#F)F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "
(a) The moment of inertia of the region about the " }{TEXT 363 1 "y" }
{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "I[y] = rho*Int(x^2*f(x),x = 0 .. 1);" "6#/&%\"IG6#%\"yG
*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"#-%\"fG6#F/F*/F/;\"\"!F*F*" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho*I
nt(2*x^2/(x+1),x = 0 .. 1);" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$*(\"\"#F'*$
%\"xGF,F',&F.F'F'F'!\"\"/F.;\"\"!F'F'" }{TEXT -1 5 " ... " }{XPPEDIT 
18 0 "PIECEWISE([u=x+1, x=u-1],[du=dx,`x =`*0*` implies  u =`*1],[``,`
x =`*1*` implies  u =`*2])" "6#-%*PIECEWISEG6%7$/%\"uG,&%\"xG\"\"\"F+F
+/F*,&F(F+F+!\"\"7$/%#duG%#dxG**%$x~=GF+\"\"!F+%.~implies~~u~=GF+F+F+7
$%!G**F4F+F+F+F6F+\"\"#F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = rho*Int(2*(u-1)^2/u,u = 1 .. 2);" "6#/%!
G*&%$rhoG\"\"\"-%$IntG6$*(\"\"#F'*$,&%\"uGF'F'!\"\"F,F'F/F0/F/;F'F,F'
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`
`=rho*Int(2*(u^2-2*u+1)/u,u=1..2)" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$*(\"
\"#F',(*$%\"uGF,F'*&F,F'F/F'!\"\"F'F'F'F/F1/F/;F'F,F'" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*Int(2*u-4+
2/u,u=1..2)" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$,(*&\"\"#F'%\"uGF'F'\"\"%!
\"\"*&F-F'F.F0F'/F.;F'F-F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = rho*(u^2-4*u+2*ln(u));" "6#/%!G*&%$rhoG
\"\"\",(*$%\"uG\"\"#F'*&\"\"%F'F*F'!\"\"*&F+F'-%#lnG6#F*F'F'F'" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIEC
EWISEG6$7$\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = rho*(-
4+2*ln(2)-(-3));" "6#/%!G*&%$rhoG\"\"\",(\"\"%!\"\"*&\"\"#F'-%#lnG6#F,
F'F',$\"\"$F*F*F'" }{XPPEDIT 18 0 "`` = (2*ln(2)-1)*rho;" "6#/%!G*&,&*
&\"\"#\"\"\"-%#lnG6#F(F)F)F)!\"\"F)%$rhoGF)" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``=(2*ln(2)
-1)*M/(2*ln(2))" "6#/%!G*(,&*&\"\"#\"\"\"-%#lnG6#F(F)F)F)!\"\"F)%\"MGF
)*&F(F)-F+6#F(F)F-" }{XPPEDIT 18 0 "``=(1-1/(2*ln(2)))*M" "6#/%!G*&,&
\"\"\"F'*&F'F'*&\"\"#F'-%#lnG6#F*F'!\"\"F.F'%\"MGF'" }{TEXT -1 1 " " }
{TEXT 392 1 "~" }{TEXT -1 14 " 0.2786524796 " }{TEXT 391 1 "M" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
50 "(a) The moment of inertia of the region about the " }{TEXT 364 1 "
x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "I[x] = rho*Int(f(x)^3/3,x = 0 .. 1);" "6#/&%\"IG6#%\"xG
*&%$rhoG\"\"\"-%$IntG6$*&-%\"fG6#F'\"\"$F2!\"\"/F';\"\"!F*F*" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3" 
"6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((
2/(x+1))^3,x = 0 .. 1);" "6#-%$IntG6$*$*&\"\"#\"\"\",&%\"xGF)F)F)!\"\"
\"\"$/F+;\"\"!F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= rho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(8/(x+1)^3,x=0..1)" "6#-%$IntG6$*&\"\")\"\"\"
*$,&%\"xGF(F(F(\"\"$!\"\"/F+;\"\"!F(" }{TEXT -1 8 " ...    " }
{XPPEDIT 18 0 "PIECEWISE([u = x+1, `x =`*0*` implies  u =`*1],[du = dx
,`x =`*1*` implies  u =`*2 ])" "6#-%*PIECEWISEG6$7$/%\"uG,&%\"xG\"\"\"
F+F+**%$x~=GF+\"\"!F+%.~implies~~u~=GF+F+F+7$/%#duG%#dxG**F-F+F+F+F/F+
\"\"#F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "``= rho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(8/u^3,u=1..2)" "6#-%$IntG6$*&\"\")\"\"\"*$%\"uG\"\"
$!\"\"/F*;F(\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= rho/3" "6#/%!G*
&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(8*u^(-1),
u=1..2)" "6#-%$IntG6$*&\"\")\"\"\")%\"uG,$F(!\"\"F(/F*;F(\"\"#" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=r
ho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"``(-4*u^(-2));" "6#-%!G6#,$*&\"\"%\"\"\")%\"uG,$\"\"#!\"\"F)F." }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIEC
EWISEG6$7$\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho/3
" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(
-4/(u^2));" "6#-%!G6#,$*&\"\"%\"\"\"*$%\"uG\"\"#!\"\"F-" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"
\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=rho/3" "6#/%!G*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "``(-1-(-4)) =1*`.`*rho" "6#/-%!G6#,&\"\"\"!\"\",$
\"\"%F)F)*(F(F(%\".GF(%$rhoGF(" }{XPPEDIT 18 0 "``=M/(2*ln(2))" "6#/%!
G*&%\"MG\"\"\"*&\"\"#F'-%#lnG6#F)F'!\"\"" }{TEXT -1 1 " " }{TEXT 398 
1 "~" }{TEXT -1 14 " 0.7213475204 " }{TEXT 399 1 "M" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Alterna
tively, the moment of inertia about the " }{TEXT 365 1 "x" }{TEXT -1 
61 " axis can be determined by using an integral with respect to " }
{TEXT 366 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "y=2/(x+1)" "6#/%\"yG*&\"\"#\"\"\",&%\"xGF'F'F'!\"\"" }
{TEXT -1 6 " when " }{XPPEDIT 18 0 "x*y+y=2" "6#/,&*&%\"xG\"\"\"%\"yGF
'F'F(F'\"\"#" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x*y=2-y" "6#/*&%\"xG
\"\"\"%\"yGF&,&\"\"#F&F'!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=2/
y-1" "6#/%\"xG,&*&\"\"#\"\"\"%\"yG!\"\"F(F(F*" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 25 "We can divide the region " }{TEXT 394 1 "
R" }{TEXT -1 41 " into a lower square and an upper region " }{XPPEDIT 
18 0 "R*`*`" "6#*&%\"RG\"\"\"%\"*GF%" }{TEXT -1 23 "  bounded by the c
urve " }{XPPEDIT 18 0 "x=2/y+1" "6#/%\"xG,&*&\"\"#\"\"\"%\"yG!\"\"F(F(
F(" }{XPPEDIT 18 0 "``=g(y)" "6#/%!G-%\"gG6#%\"yG" }{TEXT -1 6 ", the \+
" }{TEXT 393 1 "y" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "
y=1" "6#/%\"yG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "
The moment of inertia of the region " }{XPPEDIT 18 0 "R*`*`" "6#*&%\"R
G\"\"\"%\"*GF%" }{TEXT -1 12 "  about the " }{TEXT 397 1 "x" }{TEXT 
-1 11 " axis is:  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"I[x]*`*` = rho*Int(y^2*g(y),y = 0 .. 4);" "6#/*&&%\"IG6#%\"xG\"\"\"%
\"*GF)*&%$rhoGF)-%$IntG6$*&%\"yG\"\"#-%\"gG6#F1F)/F1;\"\"!\"\"%F)" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+
rho*Int(y^2*(2/y-1),y=1..2)" "6#/%!G*&%$rhoG\"\"\"-%$IntG6$*&%\"yG\"\"
#,&*&F-F'F,!\"\"F'F'F0F'/F,;F'F-F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*Int(2*y-y^2,y=1..2)" "6#/%!G*&
%$rhoG\"\"\"-%$IntG6$,&*&\"\"#F'%\"yGF'F'*$F.F-!\"\"/F.;F'F-F'" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=r
ho*(y^2-y^3/3)" "6#/%!G*&%$rhoG\"\"\",&*$%\"yG\"\"#F'*&F*\"\"$F-!\"\"F
.F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``])" "6#-%
*PIECEWISEG6$7$\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=rho*(``(4-8/3)-(1-1/3))" "6#/%!G*&
%$rhoG\"\"\",&-F$6#,&\"\"%F'*&\"\")F'\"\"$!\"\"F0F',&F'F'*&F'F'F/F0F0F
0F'" }{XPPEDIT 18 0 "``=rho*(4/3-2/3)" "6#/%!G*&%$rhoG\"\"\",&*&\"\"%F
'\"\"$!\"\"F'*&\"\"#F'F+F,F,F'" }{XPPEDIT 18 0 "``=2*rho/3" "6#/%!G*(
\"\"#\"\"\"%$rhoGF'\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The moment of inertia of \+
the lower unit square about the " }{TEXT 395 1 "y" }{TEXT -1 9 " axis \+
is " }{XPPEDIT 18 0 "rho/3" "6#*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 42 "Hence the moment of inertia of the r
egion " }{TEXT 396 1 "R" }{TEXT -1 11 " about the " }{TEXT 400 1 "x" }
{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "I[x]= 2*rho/3+rho/3" "6#/&%\"IG6#%\"xG,&*(\"\"#\"\"\"%$rhoGF+\"
\"$!\"\"F+*&F,F+F-F.F+" }{XPPEDIT 18 0 "``=1*`.`*rho" "6#/%!G*(\"\"\"F
&%\".GF&%$rhoGF&" }{XPPEDIT 18 0 "``=M/(2*ln(2))" "6#/%!G*&%\"MG\"\"\"
*&\"\"#F'-%#lnG6#F)F'!\"\"" }{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" {TEXT 
-1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 189 "f := x -> 2/(x+1);\na := 0: b := 1:\nInt(f(
x),x=a..b);\nA := value(%);\nrho*Int(x^2*f(x),x=a..b);\nIy := value(%)
:\nIy = Iy*M/(A*rho);\nrho/3*Int(f(x)^3,x=a..b);\nIx := value(%):\nIx \+
= Ix*M/(A*rho);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6
$%)operatorG%&arrowGF(,$*&\"\"#\"\"\",&9$F/F/F/!\"\"F/F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#\"\"\",&%\"xGF)F)F)!\"\"F)
/F+;\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*&\"\"#\"\"\"-%
#lnG6#F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%$rhoG\"\"\"-%$IntG6
$,$*(\"\"#F%%\"xGF+,&F,F%F%F%!\"\"F%/F,;\"\"!F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%$rhoG\"\"\",&F&!\"\"*&\"\"#F&-%#lnG6#F*F&F&F&,$*&#F
&F*F&*(F'F&%\"MGF&F+F(F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"
\"\"\"\"$F&*&%$rhoGF&-%$IntG6$,$*&\"\")F&,&%\"xGF&F&F&!\"$F&/F1;\"\"!F
&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$rhoG,$*&#\"\"\"\"\"#F(*&
%\"MGF(-%#lnG6#F)!\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The perpen
dicular axis theorem " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 110 "Let OX, OY and OZ be three mutually pe
rpendicular axes, and consider a lamina lying in the plane of OX and O
Y." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then  " }{XPPEDIT 18 0 "I[OX]+I[Oy]=
I[OZ]" "6#/,&&%\"IG6#%#OXG\"\"\"&F&6#%#OyGF)&F&6#%#OZG" }{TEXT -1 2 ".
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 427 283 283 
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l7$$\"+uXXN\"*F]_l$!+=KoL?F]_l7$$\"+g^c5&*F]_l$!+w\\3X:F]_l7$$\"+0gZ\"
y*F]_l$!+e%e&R5F]_l7$$\"+`*=_%**F]_l$!+:KUE_F`_l7$F($!+&QKz*eF2-%&COLO
RG6&Fd^l$\"#&)!\"#Feam$\"#$*Fgam-F$6&7$Fi^l7$$\"#8FjtF+7%7$$\"++++h7!
\"*$\"+++++NF`_lF]bm7$Fbbm$!+++++NF`_l-%&STYLEG6#%,PATCHNOGRIDGFa^l-F$
6&7$Fi^l7$F+F(7%7$$!+++++DF`_l$\"+++++&*F]_lFacm7$$Fd`lF`_lFfcmFjbmFa^
l-F$6&7$Fi^l7$$!\")FjtF^dm7%7$$!+0LA.vF]_l$!+&pwn&yF]_lF]dm7$FddmFbdmF
jbmFa^l-F$6%7%7$$!3++++++++DF/F[em7$$\"3w*************\\$F/F[em7$$\"3w
**************fF/F+Fa^l-%*LINESTYLEG6#\"\"#-F$6%7$Fi^lF]emFa^lFcem-F$6
&7#F]em-%'SYMBOLG6#%'CIRCLEGFa^l-F[cm6#%&POINTG-F$6&F\\fm-F^fm6#%(DIAM
ONDGFa^lFafm-F$6&F\\fm-F^fm6#%&CROSSGFa^lFafm-%%TEXTG6&7$$F_dmFgam$Ffc
lFgamQ\"O6\"Fa^l-%%FONTG6$%*HELVETICAG\"#5-F_gm6&7$$\"$B\"Fgam$FjtFjtQ
\"YFegmFa^lFfgm-F_gm6&7$$!#&)FgamF^dmQ\"XFegmFa^lFfgm-F_gm6&7$Fbgm$\"#
(*FgamQ\"ZFegmFa^lFfgm-F_gm6&7$$\"#TFgam$!#JFgamQ\"PFegmFa^lFfgm-F_gm6
&7$$\"\"%Fgam$!#LFgamQ\"yFegmFa^lFfgm-F_gm6&7$$\"#aFgam$!#9FgamQ\"xFeg
mFa^lFfgm-F_gm6&7$$\"#BFgamFbgmQ\"rFegmFa^lFfgm-F_gm6&7$$\"\")Fgam$!#O
FgamQ\"1FegmFa^l-Fggm6$FigmF`[n-F_gm6&7$$\"#dFgam$!#<FgamFc[nFa^lFd[n-
F_gm6&7$$\"#EFgam$F`_lFgamFc[nFa^lFd[n-%+AXESLABELSG6%Q!FegmFf\\n-Fggm
6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$Fi\\nFi\\n" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" 
"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Consi
der a collection of particles with masses " }{XPPEDIT 18 0 "m[1],m[2],
` . . . `,m[N]" "6&&%\"mG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"NG" }
{TEXT -1 34 "  in the plane of OX and OY where " }{XPPEDIT 18 0 "m[1]
" "6#&%\"mG6#\"\"\"" }{TEXT -1 9 " lies at " }{XPPEDIT 18 0 "P(x[1],y[
1]);" "6#-%\"PG6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 8 " lies at" }{XPPEDIT 18 0 "`
`(x[2],y[2])" "6#-%!G6$&%\"xG6#\"\"#&%\"yG6#F)" }{TEXT -1 7 ", etc. " 
}}{PARA 0 "" 0 "" {TEXT -1 66 "The moment of inertia of the collection
 of particles about OY is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "I[OY] = m[1]*x[1]^2+m[2]*x[2]^2+` . . . `+m[N]*x[N]^2;
" "6#/&%\"IG6#%#OYG,**&&%\"mG6#\"\"\"F-*$&%\"xG6#F-\"\"#F-F-*&&F+6#F2F
-*$&F06#F2F2F-F-%(~.~.~.~GF-*&&F+6#%\"NGF-*$&F06#F=F2F-F-" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "and the moment of inertia of the \+
collection of particles about OX is: " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "I[OX] = m[1]*y[1]^2+m[2]*y[2]^2+` . . . `+m[N]*y[N
]^2;" "6#/&%\"IG6#%#OXG,**&&%\"mG6#\"\"\"F-*$&%\"yG6#F-\"\"#F-F-*&&F+6
#F2F-*$&F06#F2F2F-F-%(~.~.~.~GF-*&&F+6#%\"NGF-*$&F06#F=F2F-F-" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "I[OY]+I[OX] = m[1]*(x[1]^2+y[1]^2)+m[2
]*(x[2]^2+y[2]^2)+` . . . `+m[N]*(x[N]^2+y[N]^2);" "6#/,&&%\"IG6#%#OYG
\"\"\"&F&6#%#OXGF),**&&%\"mG6#F)F),&*$&%\"xG6#F)\"\"#F)*$&%\"yG6#F)F7F
)F)F)*&&F06#F7F),&*$&F56#F7F7F)*$&F:6#F7F7F)F)F)%(~.~.~.~GF)*&&F06#%\"
NGF),&*$&F56#FJF7F)*$&F:6#FJF7F)F)F)" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=m[1]*r[1]^2+m[2]*r[2]^2+` . . .
 `+m[N]*r[N]^2" "6#/%!G,**&&%\"mG6#\"\"\"F**$&%\"rG6#F*\"\"#F*F**&&F(6
#F/F**$&F-6#F/F/F*F*%(~.~.~.~GF**&&F(6#%\"NGF**$&F-6#F:F/F*F*" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``=I[OZ]" "6#/%!G&%\"IG6#%
#OZG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "r[1],r[2],` . . . `,r[N]" "6&&%\"rG6#\"\"\"&F$6#\"\"#%(
~.~.~.~G&F$6#%\"NG" }{TEXT -1 45 "  are the distances of the respectiv
e masses " }{XPPEDIT 18 0 "m[1],m[2],` . . . `,m[N]" "6&&%\"mG6#\"\"\"
&F$6#\"\"#%(~.~.~.~G&F$6#%\"NG" }{TEXT -1 57 "  from the origin O and \+
therefore also from the axis OZ. " }}{PARA 0 "" 0 "" {TEXT -1 74 "This
 proves the perpendicular axis theorem for a collection of particles. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 275 "The \+
perpendicular axis theorem holds for a lamina lying in the plane of OX
 and OY. This can be seen by subdividing the lamina into small pieces \+
by means of a grid of lines parallel to the axes OX and OY, and then p
assing to a limit by considering progressively finer grids.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "For exam
ple, we can use the perpendicular axis theorem to find the moment of i
nertia of a disc lamina of radius " }{TEXT 447 1 "R" }{TEXT -1 19 " ab
out a diameter. " }}{PARA 0 "" 0 "" {TEXT -1 99 "If O is the centre of
 the disc and OX and OY are perpendicular axes in the plane of the dis
c, then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[OX]=I[O
Y]" "6#/&%\"IG6#%#OXG&F%6#%#OYG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 
"I[OX]+I[OY]=M*R^2/2" "6#/,&&%\"IG6#%#OXG\"\"\"&F&6#%#OYGF)*(%\"MGF)*$
%\"RG\"\"#F)F1!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "He
nce" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[OX]=I[OY]" "
6#/&%\"IG6#%#OXG&F%6#%#OYG" }{XPPEDIT 18 0 "``=M*R^2/4" "6#/%!G*(%\"MG
\"\"\"*$%\"RG\"\"#F'\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "The parallel axi
s theorem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 69 "If GG ' is an axis through the centre of gravity o
f a lamina of mass " }{TEXT 455 1 "M" }{TEXT -1 75 ", and let HH ' be \+
a parallel axis in the plane of the lamina at a distance " }{TEXT 446 
1 "h" }{TEXT -1 12 " from GG '. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I[HH*`'`] = I[GG*
`'`]+M*h^2" "6#/&%\"IG6#*&%#HHG\"\"\"%\"'GF),&&F%6#*&%#GGGF)F*F)F)*&%
\"MGF)*$%\"hG\"\"#F)F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{GLPLOT2D 302 351 351 {PLOTDATA 2 "69-%'CURVESG6$7in7$$\"\"\"\"
\"!$F*F*7$$\"3A+++a*=_%**!#=$\"3/+++KYGX5F/7$$\"3a*****p+w9y*F/$\"3#**
****z!p6z?F/7$$\"3))*****H;l0^*F/$\"39+++V*p,4$F/7$$\"3_*****pdaa8*F/$
\"35+++IkOnSF/7$$\"3Q+++QSDg')F/$\"3#*******)*******\\F/7$$\"3m*****R%
*p,4)F/$\"3'******>__y(eF/7$$\"3[*****fD[9V(F/$\"3')*****H118p'F/7$$\"
3-+++lgI\"p'F/$\"3U+++a#[9V(F/7$$\"39+++CD&y(eF/$\"3e*****H%*p,4)F/7$$
\"3;+++-+++]F/$\"3I+++PSDg')F/7$$\"3')*****RVmt1%F/$\"3Z+++vXXN\"*F/7$
$\"3++++[*p,4$F/$\"3z*****>;l0^*F/7$$\"3')*****R\"p6z?F/$\"3c+++1gZ\"y
*F/7$$\"31+++RYGX5F/$\"39+++`*=_%**F/7$$\"3y*****>>m*[z!#FF(7$$!3%****
**Rj%GX5F/F-7$$!3#******z!p6z?F/F37$$!31+++U*p,4$F/F87$$!3#******zUmt1
%F/$\"3g*****zdaa8*F/7$$!3u*****p*******\\F/$\"3a+++SSDg')F/7$$!3s****
**=D&y(eF/$\"3#******p%*p,4)F/7$$!3_******egI\"p'F/$\"3t******e#[9V(F/
7$$!34+++]#[9V(F/$\"3O+++pgI\"p'F/7$$!3O+++R*p,4)F/$\"3_******HD&y(eF/
7$$!3Q+++QSDg')F/$\"3++++++++]F/7$$!3b+++wXXN\"*F/$\"3E+++KkOnSF/7$$!3
z*****>;l0^*F/$\"3%)*****f%*p,4$F/7$$!3a*****p+w9y*F/$\"3)******>\"p6z
?F/7$$!39+++`*=_%**F/$\"3/+++PYGX5F/7$$!\"\"F*$\"3I+++&QKz*eF`p7$$!3A+
++a*=_%**F/$!3'******fi%GX5F/7$$!3q******3gZ\"y*F/$!35++++p6z?F/7$$!38
+++m^c5&*F/$!3/+++N*p,4$F/7$$!3&)*****4eaa8*F/$!3!******4Umt1%F/7$$!3w
*****R/a-m)F/$!3G+++!*******\\F/7$$!3%)*****f%*p,4)F/$!3!)******>D&y(e
F/7$$!3k*****zD[9V(F/$!3g******fgI\"p'F/7$$!3E+++ogI\"p'F/$!3<+++^#[9V
(F/7$$!3Y+++GD&y(eF/$!3W+++S*p,4)F/7$$!3Z*****p++++&F/$!3/+++MSDg')F/7
$$!3s******QkOnSF/$!3I+++tXXN\"*F/7$$!3%******R&*p,4$F/$!3j******f^c5&
*F/7$$!3++++>p6z?F/$!3[+++0gZ\"y*F/7$$!3-+++XYGX5F/$!31+++_*=_%**F/7$$
!3#)*****zd)*o%QF`pFbt7$$\"3++++GYGX5F/Fgt7$$\"3)******>!p6z?F/F\\u7$$
\"3?+++P*p,4$F/$!3/+++l^c5&*F/7$$\"31+++BkOnSF/$!3x******zXXN\"*F/7$$
\"3!)*****4*******\\F/$!3o*****H/a-m)F/7$$\"3U+++9D&y(eF/$!3;+++]*p,4)
F/7$$\"3I+++bgI\"p'F/$!31+++j#[9V(F/7$$\"3w*****fC[9V(F/$!3m*****R218p
'F/7$$\"3/+++N*p,4)F/$!3%******\\`_y(eF/7$$\"3s******HSDg')F/$!30+++9+
++]F/7$$\"3Q+++uXXN\"*F/$!37+++PkOnSF/7$$\"3j******f^c5&*F/$!3y*****>&
*p,4$F/7$$\"3[+++0gZ\"y*F/$!37+++<p6z?F/7$F[p$!3)******Hk%GX5F/7$F($!3
1+++xkez6!#E-%'COLOURG6&%$RGBGF*F*F*-%)POLYGONSG6%7jn7$F+F+F'7$$\"+a*=
_%**!#5$\"+KYGX5F_^l7$$\"+2gZ\"y*F_^l$\"+3p6z?F_^l7$$\"+j^c5&*F_^l$\"+
V*p,4$F_^l7$$\"+xXXN\"*F_^l$\"+IkOnSF_^l7$$\"+QSDg')F_^l$\"+********\\
F_^l7$$\"+W*p,4)F_^l$\"+AD&y(eF_^l7$$\"+c#[9V(F_^l$\"+jgI\"p'F_^l7$$\"
+lgI\"p'F_^l$\"+a#[9V(F_^l7$$\"+CD&y(eF_^l$\"+V*p,4)F_^l7$$\"+-+++]F_^
l$\"+PSDg')F_^l7$$\"+MkOnSF_^l$\"+vXXN\"*F_^l7$$\"+[*p,4$F_^l$\"+i^c5&
*F_^l7$$\"+9p6z?F_^l$\"+1gZ\"y*F_^l7$$\"+RYGX5F_^l$\"+`*=_%**F_^l7$$\"
+#>m*[z!#>F(7$$!+MYGX5F_^lF]^l7$$!+3p6z?F_^lFc^l7$$!+U*p,4$F_^lFh^l7$$
!+GkOnSF_^l$\"+yXXN\"*F_^l7$$!+(*******\\F_^l$\"+SSDg')F_^l7$$!+>D&y(e
F_^l$\"+Z*p,4)F_^l7$$!+fgI\"p'F_^l$\"+f#[9V(F_^l7$$!+]#[9V(F_^l$\"+pgI
\"p'F_^l7$$!+R*p,4)F_^l$\"+ID&y(eF_^l7$$!+QSDg')F_^l$\"+++++]F_^l7$$!+
wXXN\"*F_^l$\"+KkOnSF_^l7$$!+i^c5&*F_^l$\"+Y*p,4$F_^l7$$!+2gZ\"y*F_^l$
\"+7p6z?F_^l7$$!+`*=_%**F_^l$\"+PYGX5F_^l7$Fbt$\"+&QKz*eFfbl7$$!+a*=_%
**F_^l$!+EYGX5F_^l7$$!+4gZ\"y*F_^l$!++p6z?F_^l7$$!+m^c5&*F_^l$!+N*p,4$
F_^l7$$!+\"eaa8*F_^l$!+@kOnSF_^l7$$!+WSDg')F_^l$!+!*******\\F_^l7$$!+Y
*p,4)F_^l$!+?D&y(eF_^l7$$!+e#[9V(F_^l$!+ggI\"p'F_^l7$$!+ogI\"p'F_^l$!+
^#[9V(F_^l7$$!+GD&y(eF_^l$!+S*p,4)F_^l7$$!+2+++]F_^l$!+MSDg')F_^l7$$!+
RkOnSF_^l$!+tXXN\"*F_^l7$$!+a*p,4$F_^l$!+g^c5&*F_^l7$$!+>p6z?F_^l$!+0g
Z\"y*F_^l7$$!+XYGX5F_^l$!+_*=_%**F_^l7$$!+y&)*o%QFfblFbt7$$\"+GYGX5F_^
lF[gl7$$\"+-p6z?F_^lF`gl7$$\"+P*p,4$F_^l$!+l^c5&*F_^l7$$\"+BkOnSF_^l$!
+!eaa8*F_^l7$$\"+\"*******\\F_^l$!+VSDg')F_^l7$$\"+9D&y(eF_^l$!+]*p,4)
F_^l7$$\"+bgI\"p'F_^l$!+j#[9V(F_^l7$$\"+Y#[9V(F_^l$!+ugI\"p'F_^l7$$\"+
N*p,4)F_^l$!+ND&y(eF_^l7$$\"+ISDg')F_^l$!+9+++]F_^l7$$\"+uXXN\"*F_^l$!
+PkOnSF_^l7$$\"+g^c5&*F_^l$!+_*p,4$F_^l7$$\"+0gZ\"y*F_^l$!+<p6z?F_^l7$
Fabl$!+VYGX5F_^l7$F($!+xkez6F/-%&COLORG6&Ff]l$\"#&)!\"#Fi_m$\"#$*F[`m-
%&STYLEG6#%,PATCHNOGRIDG-F$6$7$7$F+$!3%**************>\"!#<7$F+$\"3%**
************>\"Fh`mFc]l-F$6$7$7$$\"3a**************pF/Ff`m7$F`amFj`mFc
]l-F$6%7$7$F+$\"3C+++++++MF/7$F`amFgamFc]l-%*LINESTYLEG6#\"\"#-F$6&7#7
$$\"3A+++++++SF/Fgam-%'SYMBOLG6#%'CIRCLEGFc]l-F_`m6#%&POINTG-F$6&F`bm-
Febm6#%(DIAMONDGFc]lFhbm-F$6&F`bm-Febm6#%&CROSSGFc]lFhbm-F$6&7#F[^lFc]
lFdbmFhbm-F$6&7$7$$\"#VF[`m$!#XF[`m7$$\"#qF[`mF^dm7%7$$\"++++\\mF_^l$!
++++]TF_^lF`dm7$Fedm$!++++][F_^lF^`mFc]l-F$6&7$7$$\"#FF[`mF^dm7$F+F^dm
7%7$$\"*+++^$F_^lFjdmFbem7$FeemFgdmF^`mFc]l-%%TEXTG6&7$$!\")F[`m$!$B\"
F[`mQ\"G6\"Fc]l-%%FONTG6$%*HELVETICAG\"#5-Fiem6&7$$!\"*F[`m$\"#7FctQ#G
'FafmFc]lFbfm-Fiem6&7$$\"#yF[`mF^fmQ\"HFafmFc]lFbfm-Fiem6&7$FbgmF\\gmQ
#H'FafmFc]lFbfm-Fiem6&7$$F]bmFct$\"#EF[`mQ\"xFafmFc]lFbfm-Fiem6&7$$\"#
NF[`mF^dmQ\"hFafmFc]lFbfm-Fiem6&7$$\"\"%Fct$\"#ZF[`mQ\"mFafmFc]lFbfm-F
iem6&7$$\"#CF[`m$\"#AF[`mQ\"1FafmFc]l-Fcfm6$Fefm\"\")-Fiem6&7$F[imF\\d
mFeimFc]lFfim-%+AXESLABELSG6%Q!FafmF_jm-Fcfm6#%(DEFAULTG-%*AXESSTYLEG6
#%%NONEG-%%VIEWG6$FbjmFbjm" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" }}{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 47 "Consi
der a collection of particles with masses " }{XPPEDIT 18 0 "m[1],m[2],
` . . . `,m[N]" "6&&%\"mG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"NG" }
{TEXT -1 47 "  in the plane of the two parallel axes, where " }
{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\"\"\"" }{TEXT -1 20 " lies at a dista
nce " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 13 " from GG \+
`,  " }{XPPEDIT 18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 20 " lies at a
 distance " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" }{TEXT -1 16 " fro
m GG ' etc. " }}{PARA 0 "" 0 "" {TEXT -1 68 "The moment of inertia of \+
the collection of particles about GG ' is: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "I[GG*`'`] = m[1]*x[1]^2+m[2]*x[2]^2+` .
 . . `+m[N]*x[N]^2;" "6#/&%\"IG6#*&%#GGG\"\"\"%\"'GF),**&&%\"mG6#F)F)*
$&%\"xG6#F)\"\"#F)F)*&&F.6#F4F)*$&F26#F4F4F)F)%(~.~.~.~GF)*&&F.6#%\"NG
F)*$&F26#F?F4F)F)" }{XPPEDIT 18 0 "``=Sum(m[i]*x[i]^2,i=1..N)" "6#/%!G
-%$SumG6$*&&%\"mG6#%\"iG\"\"\"*$&%\"xG6#F,\"\"#F-/F,;F-%\"NG" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "I[HH*`'`] =m[1]*(h-x[1])^2+m[2]*(h-x[2]
)^2+` . . . `+m[N]*(h-x[N])^2" "6#/&%\"IG6#*&%#HHG\"\"\"%\"'GF),**&&%
\"mG6#F)F)*$,&%\"hGF)&%\"xG6#F)!\"\"\"\"#F)F)*&&F.6#F7F)*$,&F2F)&F46#F
7F6F7F)F)%(~.~.~.~GF)*&&F.6#%\"NGF)*$,&F2F)&F46#FCF6F7F)F)" }{XPPEDIT 
18 0 "``= Sum(m[i]*(h-x[i])^2,i = 1 .. N)" "6#/%!G-%$SumG6$*&&%\"mG6#%
\"iG\"\"\"*$,&%\"hGF-&%\"xG6#F,!\"\"\"\"#F-/F,;F-%\"NG" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(``(m[i]*
h^2-2*h*m[i]*x[i]+m[i]*x[i]^2),i = 1 .. N);" "6#/%!G-%$SumG6$-F$6#,(*&
&%\"mG6#%\"iG\"\"\"*$%\"hG\"\"#F0F0**F3F0F2F0&F-6#F/F0&%\"xG6#F/F0!\"
\"*&&F-6#F/F0*$&F86#F/F3F0F0/F/;F0%\"NG" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = h^2*Sum(m[i],i = 1 .. N)-2*
h*Sum(m[i]*x[i],i = 1 .. N)+Sum(m[i]*x[i]^2,i = 1 .. N);" "6#/%!G,(*&%
\"hG\"\"#-%$SumG6$&%\"mG6#%\"iG/F/;\"\"\"%\"NGF2F2*(F(F2F'F2-F*6$*&&F-
6#F/F2&%\"xG6#F/F2/F/;F2F3F2!\"\"-F*6$*&&F-6#F/F2*$&F;6#F/F(F2/F/;F2F3
F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=M*h^2+I[GG*`'`]" "6#/%!G,&*&%\"MG
\"\"\"*$%\"hG\"\"#F(F(&%\"IG6#*&%#GGGF(%\"'GF(F(" }{TEXT -1 3 "   " }}
{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "Sum(m[i]*x[i],i = \+
1 .. N)" "6#-%$SumG6$*&&%\"mG6#%\"iG\"\"\"&%\"xG6#F*F+/F*;F+%\"NG" }
{TEXT -1 68 " is the first moment of mass about GG ' and is therfore e
qual to 0. " }}{PARA 0 "" 0 "" {TEXT -1 294 "The parallel axis theorem
 holds for a lamina lying in the plane of GG ' and HH '. This can be s
een by subdividing the lamina into small pieces by means of a grid of \+
lines parallel and perpendicular to the axes GG ' and HH ', and then p
assing to a limit by considering progressively finer grids." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "For example, w
e can use the perpendicular axis theorem to find the moment of inertia
 of a disc lamina of radius " }{TEXT 448 1 "R" }{TEXT -1 10 " and mass
 " }{TEXT 449 1 "M" }{TEXT -1 18 " about a tangent. " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{GLPLOT2D 198 198 198 {PLOTDATA 2 "61-%'CURVESG6$
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eF_^l$!+]*p,4)F_^l7$$\"+bgI\"p'F_^l$!+j#[9V(F_^l7$$\"+Y#[9V(F_^l$!+ugI
\"p'F_^l7$$\"+N*p,4)F_^l$!+ND&y(eF_^l7$$\"+ISDg')F_^l$!+9+++]F_^l7$$\"
+uXXN\"*F_^l$!+PkOnSF_^l7$$\"+g^c5&*F_^l$!+_*p,4$F_^l7$$\"+0gZ\"y*F_^l
$!+<p6z?F_^l7$Fabl$!+VYGX5F_^l7$F($!+xkez6F/-%&COLORG6&Ff]l$\"#&)!\"#F
i_m$\"#$*F[`m-%&STYLEG6#%,PATCHNOGRIDG-F$6$7$7$F+$!3%**************>\"
!#<7$F+$\"3%**************>\"Fh`mFc]l-F$6$7$7$F(Ff`m7$F(Fj`mFc]l-F$6&7
#F[^lFc]l-%'SYMBOLG6#%'CIRCLEG-F_`m6#%&POINTG-F$6&7$7$$\"#dF[`m$!#XF[`
m7$$\"$+\"F[`mFabm7%7$$\"++++T%*F_^l$!++++]TF_^lFcbm7$Fhbm$!++++][F_^l
F^`mFc]l-F$6&7$7$$\"#VF[`mFabm7$F+Fabm7%7$$\"*+++f&F_^lF]cmFecm7$FhcmF
jbmF^`mFc]l-%%TEXTG6&7$$FctFct$!$B\"F[`mQ\"G6\"Fc]l-%%FONTG6$%*HELVETI
CAG\"#5-F\\dm6&7$F_dm$\"#7FctQ#G'FcdmFc]lFddm-F\\dm6&7$$\"#6FctF`dmQ\"
AFcdmFc]lFddm-F\\dm6&7$FbemF\\emQ#A'FcdmFc]lFddm-F\\dm6&7$$\"\"&FctFab
mQ\"RFcdmFc]l-Fedm6%%&TIMESG%'ITALICGFhdm-%+AXESLABELSG6%Q!FcdmFffm-Fe
dm6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FifmFifm" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
43 "The centre of gravity is at the centre, so " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "I[AA*`'`]=I[GG*`'`]+M*R^2" "6#/&%\"IG6#
*&%#AAG\"\"\"%\"'GF),&&F%6#*&%#GGGF)F*F)F)*&%\"MGF)*$%\"RG\"\"#F)F)" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=M
*R^2/4+M*R^2" "6#/%!G,&*(%\"MG\"\"\"*$%\"RG\"\"#F(\"\"%!\"\"F(*&F'F(*$
F*F+F(F(" }{XPPEDIT 18 0 "``=5/4*M*R^2" "6#/%!G**\"\"&\"\"\"\"\"%!\"\"
%\"MGF'%\"RG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "T
asks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 123 "Calculate t
he radius of gyration of a rectangular door 2 m high and 1.2 m wide ab
out a vertical axis through its hinges.   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "1.2/sqrt(3)" "6#*&-%&FloatG6$\"#7!\"\"
\"\"\"-%%sqrtG6#\"\"$F(" }{TEXT -1 1 " " }{TEXT 450 1 "~" }{TEXT -1 
12 " 0.69282 m. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 
"Q2 " }}{PARA 0 "" 0 "" {TEXT -1 61 "Find the moment of inertia of a r
ectangular lamina of length " }{TEXT 451 1 "a" }{TEXT -1 8 ", width " 
}{TEXT 452 1 "b" }{TEXT -1 10 " and mass " }{TEXT 453 1 "M" }{TEXT -1 
82 " about an axis perpendicular to the plane of the lamina and throug
h its centre.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "M/3" "6#*&%\"MG\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "``(a^2+b^2)" "6
#-%!G6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF)F*" }{TEXT -1 2 "  " }}}{PARA 0 "
" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{TEXT 408 1 "R" }{TEXT -1 71 " the plane triangular
 region in the first quadrant bounded by the line " }{XPPEDIT 18 0 "y \+
= 2-x;" "6#/%\"yG,&\"\"#\"\"\"%\"xG!\"\"" }{TEXT -1 9 " and the " }
{TEXT 406 1 "x" }{TEXT -1 5 " and " }{TEXT 407 1 "y" }{TEXT -1 6 " axe
s." }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the moment of inertia of a uni
form lamima in the shape of the region " }{TEXT 411 1 "R" }{TEXT -1 
15 " (a) about the " }{TEXT 410 1 "y" }{TEXT -1 20 " axis (b) about th
e " }{TEXT 409 1 "x" }{TEXT -1 28 " axis, in terms of the mass " }
{TEXT 415 1 "M" }{TEXT -1 17 " of the lamina.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "
" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "I[y]=rho*Int(x^2*(2-x),x = 0
 .. 2)" "6#/&%\"IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"#,&F0F*F/!
\"\"F*/F/;\"\"!F0F*" }{XPPEDIT 18 0 "``=4/3*rho" "6#/%!G*(\"\"%\"\"\"
\"\"$!\"\"%$rhoGF'" }{XPPEDIT 18 0 " ``= 2/3*M" "6#/%!G*(\"\"#\"\"\"\"
\"$!\"\"%\"MGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " (b) " 
}{XPPEDIT 18 0 "I[x]=rho/3" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"\"\"$!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2-x)^3,x = 0 .. 2)=4/3*rho" "6#
/-%$IntG6$*$,&\"\"#\"\"\"%\"xG!\"\"\"\"$/F+;\"\"!F)*(\"\"%F*F-F,%$rhoG
F*" }{XPPEDIT 18 0 " ``= 2/3*M" "6#/%!G*(\"\"#\"\"\"\"\"$!\"\"%\"MGF'
" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "rho = M/2;" "6#/%$rhoG*&%\"MG
\"\"\"\"\"#!\"\"" }{TEXT -1 16 " is the density." }}}{PARA 0 "" 0 "" 
{TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________
_________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 4 
"Let " }{TEXT 419 1 "R" }{TEXT -1 61 " the plane region in the first q
uadrant bounded by the curve " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%
\"xG\"\"#" }{TEXT -1 11 ", the line " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"
\"$" }{TEXT -1 9 " and the " }{TEXT 417 1 "x" }{TEXT -1 5 " and " }
{TEXT 418 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 74 "Fi
nd the moment of inertia of a uniform lamima in the shape of the regio
n " }{TEXT 422 1 "R" }{TEXT -1 15 " (a) about the " }{TEXT 421 1 "y" }
{TEXT -1 20 " axis (b) about the " }{TEXT 420 1 "x" }{TEXT -1 28 " axi
s, in terms of the mass " }{TEXT 423 1 "M" }{TEXT -1 17 " of the lamin
a.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "I[y]=
rho*Int(x^4,x = 0 .. 3)" "6#/&%\"IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*$%\"
xG\"\"%/F/;\"\"!\"\"$F*" }{XPPEDIT 18 0 "``=243/5*rho" "6#/%!G*(\"$V#
\"\"\"\"\"&!\"\"%$rhoGF'" }{XPPEDIT 18 0 "``= 27/5*M" "6#/%!G*(\"#F\"
\"\"\"\"&!\"\"%\"MGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "
 (b) " }{XPPEDIT 18 0 "I[x]=rho/3" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"\"\"
$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " Int(x^6,x = 0 .. 3)=729/7*rho
" "6#/-%$IntG6$*$%\"xG\"\"'/F(;\"\"!\"\"$*(\"$H(\"\"\"\"\"(!\"\"%$rhoG
F0" }{XPPEDIT 18 0 " ``= 81/7*M" "6#/%!G*(\"#\")\"\"\"\"\"(!\"\"%\"MGF
'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "rho = M/9;" "6#/%$rhoG*&%\"M
G\"\"\"\"\"*!\"\"" }{TEXT -1 16 " is the density." }}}{PARA 0 "" 0 "" 
{TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________
_________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q5  " }}{PARA 0 "" 0 "" {TEXT -1 
4 "Let " }{TEXT 425 1 "R" }{TEXT -1 61 " the plane region in the first
 quadrant bounded by the curve " }{XPPEDIT 18 0 "y = x-x^2;" "6#/%\"yG
,&%\"xG\"\"\"*$F&\"\"#!\"\"" }{TEXT -1 9 " and the " }{TEXT 424 1 "x" 
}{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the moment of
 inertia of a uniform lamima in the shape of the region " }{TEXT 428 
1 "R" }{TEXT -1 15 " (a) about the " }{TEXT 427 1 "y" }{TEXT -1 20 " a
xis (b) about the " }{TEXT 426 1 "x" }{TEXT -1 28 " axis, in terms of \+
the mass " }{TEXT 429 1 "M" }{TEXT -1 17 " of the lamina.  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}
{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "I[y]=rho*Int(x^2*(x
-x^2),x = 0 .. 1)" "6#/&%\"IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*&%\"xG\"\"
#,&F/F**$F/F0!\"\"F*/F/;\"\"!F*F*" }{XPPEDIT 18 0 "`` = rho/20;" "6#/%
!G*&%$rhoG\"\"\"\"#?!\"\"" }{XPPEDIT 18 0 "`` = 3/10*M" "6#/%!G*(\"\"$
\"\"\"\"#5!\"\"%\"MGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 
" (b) " }{XPPEDIT 18 0 "I[x]=rho/3" "6#/&%\"IG6#%\"xG*&%$rhoG\"\"\"\"
\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-x^2)^3,x = 0 .. 1)=rh
o/420" "6#/-%$IntG6$*$,&%\"xG\"\"\"*$F)\"\"#!\"\"\"\"$/F);\"\"!F**&%$r
hoGF*\"$?%F-" }{XPPEDIT 18 0 " ``= M/70" "6#/%!G*&%\"MG\"\"\"\"#q!\"\"
" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "rho = 6*M;" "6#/%$rhoG*&\"\"'
\"\"\"%\"MGF'" }{TEXT -1 16 " is the density." }}}{PARA 0 "" 0 "" 
{TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________
_________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q6  " }}{PARA 0 "" 0 "" {TEXT -1 
4 "Let " }{TEXT 435 1 "R" }{TEXT -1 61 " the plane region in the first
 quadrant bounded by the curve " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"y
G-%$expG6#%\"xG" }{TEXT -1 11 ", the line " }{XPPEDIT 18 0 "x=1" "6#/%
\"xG\"\"\"" }{TEXT -1 9 " and the " }{TEXT 434 1 "x" }{TEXT -1 5 " and
 " }{TEXT 440 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 
74 "Find the moment of inertia of a uniform lamima in the shape of the
 region " }{TEXT 438 1 "R" }{TEXT -1 15 " (a) about the " }{TEXT 437 
1 "y" }{TEXT -1 20 " axis (b) about the " }{TEXT 436 1 "x" }{TEXT -1 
28 " axis, in terms of the mass " }{TEXT 439 1 "M" }{TEXT -1 17 " of t
he lamina.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "
" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 
0 "I[y] =rho*Int(x^2*exp(x),x = 0 .. 1)" "6#/&%\"IG6#%\"yG*&%$rhoG\"\"
\"-%$IntG6$*&%\"xG\"\"#-%$expG6#F/F*/F/;\"\"!F*F*" }{XPPEDIT 18 0 "``=
rho*(exp(1)-2)" "6#/%!G*&%$rhoG\"\"\",&-%$expG6#F'F'\"\"#!\"\"F'" }
{XPPEDIT 18 0 "``= (exp(1)-2)*M/(exp(1)-1)" "6#/%!G*(,&-%$expG6#\"\"\"
F*\"\"#!\"\"F*%\"MGF*,&-F(6#F*F*F*F,F," }{TEXT -1 2 "  " }{TEXT 441 1 
"~" }{TEXT -1 14 " 0.4180232931 " }{TEXT 442 1 "M" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 5 " (b) " }{XPPEDIT 18 0 "I[x]=rho/3" "6#/&%
\"IG6#%\"xG*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(exp(3*x),x = 0 .. 1)=rho/9*(exp(3)-1)" "6#/-%$IntG6$-%$expG6#*&\"\"
$\"\"\"%\"xGF,/F-;\"\"!F,*(%$rhoGF,\"\"*!\"\",&-F(6#F+F,F,F4F," }
{XPPEDIT 18 0 "`` =  (exp(3)-1)*M/(9*(exp(1)-1))" "6#/%!G*(,&-%$expG6#
\"\"$\"\"\"F+!\"\"F+%\"MGF+*&\"\"*F+,&-F(6#F+F+F+F,F+F," }{TEXT -1 1 "
 " }{TEXT 443 1 "~" }{TEXT -1 13 " 1.234148659 " }{TEXT 444 1 "M" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "rho = M/(exp(1)-1);" "6#/%$rhoG*
&%\"MG\"\"\",&-%$expG6#F'F'F'!\"\"F," }{TEXT -1 16 " is the density." 
}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__
___________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q7  " }}
{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 402 1 "R" }{TEXT -1 61 " the \+
plane region in the first quadrant bounded by the curve " }{XPPEDIT 
18 0 "y = 6*x/(x+1);" "6#/%\"yG*(\"\"'\"\"\"%\"xGF',&F(F'F'F'!\"\"" }
{TEXT -1 11 ", the line " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }
{TEXT -1 9 " and the " }{TEXT 401 1 "x" }{TEXT -1 6 " axis." }}{PARA 
0 "" 0 "" {TEXT -1 74 "Find the moment of inertia of a uniform lamima \+
in the shape of the region " }{TEXT 405 1 "R" }{TEXT -1 15 " (a) about
 the " }{TEXT 404 1 "y" }{TEXT -1 20 " axis (b) about the " }{TEXT 
403 1 "x" }{TEXT -1 28 " axis, in terms of the mass " }{TEXT 416 1 "M
" }{TEXT -1 17 " of the lamina.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 
5 " (a) " }{XPPEDIT 18 0 "I[y]=rho*Int(6*x^3/(x+1),x = 0 .. 1)" "6#/&%
\"IG6#%\"yG*&%$rhoG\"\"\"-%$IntG6$*(\"\"'F**$%\"xG\"\"$F*,&F1F*F*F*!\"
\"/F1;\"\"!F*F*" }{XPPEDIT 18 0 "``=rho*(5-6*ln(2))" "6#/%!G*&%$rhoG\"
\"\",&\"\"&F'*&\"\"'F'-%#lnG6#\"\"#F'!\"\"F'" }{XPPEDIT 18 0 "`` = (5-
6*ln(2))*M/(6-6*ln(2))" "6#/%!G*(,&\"\"&\"\"\"*&\"\"'F(-%#lnG6#\"\"#F(
!\"\"F(%\"MGF(,&F*F(*&F*F(-F,6#F.F(F/F/" }{TEXT -1 1 " " }{TEXT 430 1 
"~" }{TEXT -1 14 " 0.4568514411 " }{TEXT 431 1 "M" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 5 " (b) " }{XPPEDIT 18 0 "I[x]=rho/3" "6#/&%
\"IG6#%\"xG*&%$rhoG\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(216*x^3/(x+1)^3,x = 0 .. 1)=9*rho*(17-24*ln(2))" "6#/-%$IntG6$*(\"$
;#\"\"\"*$%\"xG\"\"$F)*$,&F+F)F)F)F,!\"\"/F+;\"\"!F)*(\"\"*F)%$rhoGF),
&\"#<F)*&\"#CF)-%#lnG6#\"\"#F)F/F)" }{XPPEDIT 18 0 "`` = 9*(17-24*ln(2
))*M/(6-6*ln(2));" "6#/%!G**\"\"*\"\"\",&\"#<F'*&\"#CF'-%#lnG6#\"\"#F'
!\"\"F'%\"MGF',&\"\"'F'*&F3F'-F-6#F/F'F0F0" }{TEXT -1 1 " " }{TEXT 
432 1 "~" }{TEXT -1 13 " 1.781640791 " }{TEXT 433 1 "M" }{TEXT -1 8 ",
 where " }{XPPEDIT 18 0 "rho = M/(6-6*ln(2));" "6#/%$rhoG*&%\"MG\"\"\"
,&\"\"'F'*&F)F'-%#lnG6#\"\"#F'!\"\"F/" }{TEXT -1 16 " is the density.
" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________________________
_" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "
" {TEXT -1 45 "moment of inertia of collection of particles " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1390 "
n1 := 30: n2 := 20:\nh1 := evalf(1/n1): h2 := evalf(1/n2):\nd := evalf
(Pi/n2): a := .2: b := .1:\np1 := plot([[-1,-1],[6,6]],color=black):\n
pts := [seq([1.6*(i*h2)^2-1.6*i*h2+2.8,.8*(i*h2)^2],i=0..n2),\n      s
eq([1.6*i*h2+6.2,-.8*(i*h2)^2+1.6*i*h2+4.2],i=0..n2),\n      seq([-6.5
*(i*h1)^2+8.*i*h1+10.,-2.5*(i*h1)^2+5],i=0..n1),\n      seq([-1.5*(i*h
1)^2-5.*i*h1+11.5,2.5*(i*h1-1)^2],i=0..n1),[2.8,0]]:\np2 := plot(pts,c
olor=brown):\np3 := plots[polygonplot](pts,color=COLOR(RGB,.9,.8,.7)):
\nd1 := plots[polygonplot]([seq([8.4+a*cos(i*d),4.2+b*sin(i*d)],i=0..2
*n2)],color=brown):\nd2 := plots[polygonplot]([seq([10.1+a*cos(i*d),2.
7+b*sin(i*d)],i=0..2*n2)],color=brown):\nd3 := plots[polygonplot]([seq
([5.9+a*cos(i*d),1.5+b*sin(i*d)],i=0..2*n2)],color=brown):\np4 := plot
([[[1.5,1.5],[5.7,1.5]],[[2.7,2.7],[9.9,2.7]],[[4.2,4.2],[8.2,4.2]]],
\n  linestyle=3,color=COLOR(RGB,0,.6,0)):\nt1 := plots[textplot]([[5.6
,4.6,`r`],[6.9,2.4,`r`],\n  [4,1.2,`r`]],font=[HELVETICA,10],color=COL
OR(RGB,0,.6,0)):\nt2 := plots[textplot]([[5.75,4.5,`1`],[7.05,2.3,`2`]
,\n  [4.15,1.1,`3`]],font=[HELVETICA,8],color=COLOR(RGB,0,.6,0)):\nt3 \+
:= plots[textplot]([[9,4.4,`m`],[10.6,3,`m`],\n[6.5,1.4,`m`],[-1.1,-1.
2,`A`],[6.2,6.1,`B`]],font=[HELVETICA,10],color=black):\nt4 := plots[t
extplot]([[9.25,4.3,`1`],[10.85,2.9,`2`],\n[6.75,1.3,`3`]],font=[HELVE
TICA,8],color=black):\nplots[display]([d1,d2,d3,p1,p2,p3,p4,t1,t2,t3,t
4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 57 "bezier([[2.8,0],[2,0],[2.8,.8]],animate=false,
info=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bezier([[6.2,4.2],[7
,5],[7.8,5]],animate=false,info=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "bezier([[10,5],[14,5],[11.5,2.5]],animate=false,info=true):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "bezier([[11.5,2.5],[9,0],[5,0]],an
imate=false,info=true):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" 
{TEXT -1 47 "moment of inertia of a rod, disc and rectangle " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "h
 := 0.05:\np1 := plot([[[-1,-h],[-1,h],[1,h],[1,-h],[-1,-h]],\n      [
[-1,0],[-1.5,0]]],color=black):\np2 := plottools[arrow]([0,-.7],[0,.7]
,0,.05,.05,arrow,color=black):\np3 := plottools[arrow]([1,0],[1.5,0],0
,.07,.1,arrow,color=black):\np4 := plottools[arrow]([-.2,.35],[1,.35],
0,.06,.04,arrow,color=black):\np5 := plottools[arrow]([-.4,.35],[-1,.3
5],0,.06,.07,arrow,color=black):\np6 := plot([[[1,h],[1,.4]],[[-1,h],[
-1,.4]],\n     [[.46,-h],[.46,h]],[[.52,-h],[.52,h]],\n     [[.49,-h],
[.49,-2*h]]],color=black,linestyle=2):\np7 := plots[polygonplot]([[.46
,-h],[.46,h],[.52,h],[.52,-h]],\n       style=patchnogrid,color=COLOR(
RGB,.7,.7,.9)):\nt1 := plots[textplot]([[.49,-.15,`x`],[-.07,.65,`y`],
\n       [-.3,.36,'a'],[.51,.2,`x`]],color=black,font=[HELVETICA,10]):
\nt2 := plots[textplot]([.47,.2,'d'],\n          color=black,font=[SYM
BOL,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2],axes=none);" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 891 "h := 0.05:\np1 := plot([[[-
1,-h],[-1,h],[1,h],[1,-h],[-1,-h]],\n      [[-1,0],[-1.5,0]]],color=bl
ack):\np2 := plottools[arrow]([-1,-.7],[-1,.7],0,.05,.05,arrow,color=b
lack):\np3 := plottools[arrow]([1,0],[1.5,0],0,.07,.1,arrow,color=blac
k):\np4 := plottools[arrow]([.1,.35],[1,.35],0,.06,.06,arrow,color=bla
ck):\np5 := plottools[arrow]([-.1,.35],[-1,.35],0,.06,.06,arrow,color=
black):\np6 := plot([[[1,h],[1,.4]],[[-1,h],[-1,.4]],\n     [[.46,-h],
[.46,h]],[[.52,-h],[.52,h]],\n     [[.49,-h],[.49,-2*h]]],color=black,
linestyle=2):\np7 := plots[polygonplot]([[.46,-h],[.46,h],[.52,h],[.52
,-h]],\n       style=patchnogrid,color=COLOR(RGB,.7,.7,.9)):\nt1 := pl
ots[textplot]([[.49,-.15,`x`],[-1.07,.65,`y`],\n       [0,.36,'a'],[.5
1,.2,`x`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([.
47,.2,'d'],\n          color=black,font=[SYMBOL,10]):\nplots[display](
[p1,p2,p3,p4,p5,p6,p7,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1269 "d := evalf(Pi/30):\nr1 := 0.7: r2 := .73: rr := 0.5
: h := 0.05:\np1 := plot([[seq([r1*cos(i*d),r1*rr*sin(i*d)],i=0..60)],
\n      [seq([r2*cos(i*d),r2*rr*sin(i*d)],i=0..60)]],color=black):\np2
 := plots[polygonplot]([seq([[r1*cos(i*d),r1*rr*sin(i*d)],\n [r1*cos((
i+1)*d),r1*rr*sin((i+1)*d)],[r2*cos((i+1)*d),r2*rr*sin((i+1)*d)],\n   \+
  [r2*cos(i*d),r2*rr*sin(i*d)]],i=0..59)],\n     color=COLOR(RGB,.73,.
73,.8),style=patchnogrid):\np3 := plot([seq([cos(i*d),rr*sin(i*d)],i=0
..60)],color=black):\np4 := plots[polygonplot]([[0,0],seq([cos(i*d),rr
*sin(i*d)],i=0..60)],\n      color=COLOR(RGB,.85,.85,.93),style=patchn
ogrid):\np5 := plot([[[0,0],[0,.7]],[[r1/2-.05,0],[0,0]]],color=black)
:\np6 := plot([[0,0],[0,-.7]],color=black,linestyle=3):\np7 := plot([[
[0,0]]$3],style=point,\n     symbol=[circle,diamond,cross],color=black
):\np8 := plottools[arrow]([r1/2+.05,0],[r1,0],0,.05,.1,arrow,color=bl
ack):\np9 := plottools[arrow]([.08-r1,0],[-r1,0],0,.04,.35,arrow,color
=black):\np10 := plottools[arrow]([-.08-r2,0],[-r2,0],0,.04,.35,arrow,
color=black):\nt1 := plots[textplot]([[r1/2,0,`r`],[-.55,.08,`r`]],\n \+
         color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([-.5
8,.08,'d'],\n          color=black,font=[SYMBOL,10]):\nplots[display](
[p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,t1,t2],axes=none);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1031 "p1 := plot([[[0,.5],[1,.5]
,[1,0]],\n           [[.7,0],[.7,.5]],[[.72,0],[.72,.5]]],color=black)
:\np2 := plottools[arrow]([0,-.1],[0,.7],0,.03,.04,arrow,color=black):
\np3 := plottools[arrow]([0,0],[1.3,0],0,.03,.02,arrow,color=black):\n
p4 := plottools[arrow]([.78,.35],[.72,.35],0,.03,.3,arrow,color=black)
:\np5 := plottools[arrow]([.64,.35],[.7,.35],0,.03,.3,arrow,color=blac
k):\np6 := plottools[arrow]([.3,.25],[0,.25],0,.03,.1,arrow,color=blac
k):\np7 := plottools[arrow]([.4,.25],[.7,.25],0,.03,.1,arrow,color=bla
ck):\np8 := plots[polygonplot]([[.7,0],[.7,.5],[.72,.5],[.72,0]],\n   \+
    style=patchnogrid,color=COLOR(RGB,.7,.7,.9)):\np9 := plots[polygon
plot]([[0,0],[0,.5],[1,.5],[1,0]],\n       style=patchnogrid,color=COL
OR(RGB,.9,.9,.95)):\nt1 := plots[textplot]([[1.26,-.05,`x`],[-.04,.65,
`y`],\n       [1,-.04,'a'],[-.03,.5,'b'],[.82,.4,`x`],[.35,.26,`x`]],
\n       color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([.8,
.4,'d'],\n          color=black,font=[SYMBOL,10]):\nplots[display]([p1
,p2,p3,p4,p5,p6,p7,p8,p9,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT -1 36 "moment of inertia of a plane region " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1491 "f := x -> 2+x^2/2:\nx1 := .5: x2 := 2.5: x3 := 1.59: x4 := 1.65:
 x5 := 1.62:\np1 := plot(f(x),x=0..3,color=red,thickness=2):\np2 := pl
ot([[[x1,0],[x1,f(x1)]],[[x2,0],[x2,f(x2)]],\n    [[x3,0],[x3,f(x3)]],
[[x4,0],[x4,f(x4)]]],color=black):\np3 := plots[polygonplot]([[x3,0],[
x3,f(x3)],[x5,f(x5)],[x4,f(x4)],[x4,0]],\n     color=COLOR(RGB,.8,.8,.
83),style=patchnogrid):\npp := plot(f(x),x=x1..x2):\np4 := plots[polyg
onplot]([[x1,0],op(op(1,op(1,pp))),[x2,0]],\n     color=COLOR(RGB,.9,.
9,.93),style=patchnogrid):\np5 := plot([[[x5,f(x5)],[2.3,f(x5)]],\n   \+
[[x5,0],[x5,-.15]]],color=black,linestyle=2):\np6 := plottools[arrow](
[2.2,f(x5)/2+.3],[2.2,f(x5)],0,.05,.08,arrow,color=black):\np7 := plot
tools[arrow]([2.2,f(x5)/2-.3],[2.2,0],0,.05,.08,arrow,color=black):\np
8 := plottools[arrow]([.9,f(x5)/2],[x5,f(x5)/2],0,.13,.08,arrow,color=
black):\np9 := plottools[arrow]([.72,f(x5)/2],[0,f(x5)/2],0,.13,.08,ar
row,color=black):\np10 := plottools[arrow]([x3-.2,.7],[x3,.7],0,.11,.3
,arrow,color=black):\np11 := plottools[arrow]([x4+.2,.7],[x4,.7],0,.11
,.3,arrow,color=black):\nt1 := plots[textplot]([3,5.9,`y = f(x)`],colo
r=red,font=[HELVETICA,10]):\nt2 := plots[textplot]([[x5,-.3,`x`],[-.1,
6.3,`y`],\n    [x1,-.15,`x = a`],[x2,-.15,`x = b`],[.81,1.7,`x`],[1.45
,.95,`x`],\n      [2.2,f(x5)/2,`f(x)`]],color=black,font=[HELVETICA,10
]):\nt3 := plots[textplot]([1.4,.95,'d'],\n          color=black,font=
[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,t1,t
2,t3],\n   tickmarks=[0,0],view=[-.2..3,-.3..6.5]);\n" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 42 "perpendicular and parallel axis t
heorems  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 968 "d := evalf(Pi/30): rr := 0.5:\np1 := plot([seq([cos(
i*d),rr*sin(i*d)],i=0..60)],color=black):\np2 := plots[polygonplot]([[
0,0],seq([cos(i*d),rr*sin(i*d)],i=0..60)],\n         color=COLOR(RGB,.
85,.85,.93),style=patchnogrid):\np3 := plottools[arrow]([0,0],[1.3,0],
0,.07,.03,arrow,color=black):\np4 := plottools[arrow]([0,0],[0,1],0,.0
5,.05,arrow,color=black):\np5 := plottools[arrow]([0,0],[-.8,-.8],0,.0
5,.04,arrow,color=black):\np6 := plot([[[-.25,-.25],[.35,-.25],[.6,0]]
,\n       [[0,0],[.35,-.25]]],color=black,linestyle=2):\np7 := plot([[
[.35,-.25]]$3],style=point,\n     symbol=[circle,diamond,cross],color=
black):\nt1 := plots[textplot]([[-.08,.05,`O`],[1.23,-.1,`Y`],[-.85,-.
8,`X`],\n      [-.08,.97,`Z`],[.41,-.31,`P`],[.04,-.33,`y`],[.54,-.14,
`x`],[.23,-.08,`r`]],\n       color=black,font=[HELVETICA,10]):\nt2 :=
 plots[textplot]([[.08,-.36,`1`],[.57,-.17,`1`],[.26,-.11,`1`]],\n    \+
   color=black,font=[HELVETICA,8]):\nplots[display]([p1,p2,p3,p4,p5,p6
,p7,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 964 "d := evalf(Pi/30):\np1 := plot([se
q([cos(i*d),sin(i*d)],i=0..60)],color=black):\np2 := plots[polygonplot
]([[0,0],seq([cos(i*d),sin(i*d)],i=0..60)],\n         color=COLOR(RGB,
.85,.85,.93),style=patchnogrid):\np3 := plot([[[0,-1.2],[0,1.2]],[[.7,
-1.2],[.7,1.2]]],color=black):\np4 := plot([[[0,.34],[.7,.34]]],color=
black,linestyle=2):\np5 := plot([[[.4,.34]]$3],style=point,\n     symb
ol=[circle,diamond,cross],color=black):\np6 := plot([[0,0]],style=poin
t,\n     symbol=circle,color=black):\np7 := plottools[arrow]([.43,-.45
],[.7,-.45],0,.07,.13,arrow,color=black):\np8 := plottools[arrow]([.27
,-.45],[0,-.45],0,.07,.13,arrow,color=black):\nt1 := plots[textplot]([
[-.08,-1.23,`G`],[-.09,1.2,`G'`],\n     [.78,-1.23,`H`],[.78,1.2,`H'`]
,[.2,.26,`x`],\n      [.35,-.45,'h'],[.4,.47,`m`]],\n       color=blac
k,font=[HELVETICA,10]):\nt2 := plots[textplot]([[.24,.22,`1`],[.47,.43
,`1`]],\n       color=black,font=[HELVETICA,8]):\nplots[display]([p1,p
2,p3,p4,p5,p6,p7,p8,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 745 "d := evalf(Pi/30):\np1 := plot([seq([cos(i*d),sin(i*
d)],i=0..60)],color=black):\np2 := plots[polygonplot]([[0,0],seq([cos(
i*d),sin(i*d)],i=0..60)],\n         color=COLOR(RGB,.85,.85,.93),style
=patchnogrid):\np3 := plot([[[0,-1.2],[0,1.2]],[[1,-1.2],[1,1.2]]],col
or=black):\np6 := plot([[0,0]],style=point,\n     symbol=circle,color=
black):\np7 := plottools[arrow]([.57,-.45],[1,-.45],0,.07,.13,arrow,co
lor=black):\np8 := plottools[arrow]([.43,-.45],[0,-.45],0,.07,.13,arro
w,color=black):\nt1 := plots[textplot]([[-.1,-1.23,`G`],[-.1,1.2,`G'`]
,\n     [1.1,-1.23,`A`],[1.1,1.2,`A'`]],\n       color=black,font=[HEL
VETICA,10]):\nt2 := plots[textplot]([.5,-.45,R],\n       color=black,f
ont=[TIMES,ITALIC,10]):\nplots[display]([p1,p2,p3,p6,p7,p8,t1,t2],axes
=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }