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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 1210-2" }}{PARA 257 "" 0 "
" {TEXT -1 21 "Numerical Integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 96 "Here are two little subroutines to appr
oximate integrals using the Trapezoid and Parabolic Rules" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 426 "Trapezoid:=proc(f,a,b,n)\n  #f is the func
tion, a,b are the endpoints,\n  #n is the number of subintervals\nloca
l\n   Trap,  #current Trapezoid appprox\n   x,     #current x-value\n \+
  dx,    #subinterval width\n   i;     #index\n\nTrap:=evalf(f(a)+f(b)
): #initialize\nx:=a:   #start at left endpoint\ndx:=evalf((b-a)/n);  \+
#subinterval length\nfor i from 1 to n-1 do\n  x:=x+dx:\n  Trap:=Trap+
2*f(x):\nod:\nTrap:=Trap*dx/2:\nreturn(Trap);\nend:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "Simpson:=proc(f,a,b,n)\n  #f is th
e function, a,b are the endpoints,\n  #n is the number of (even) subin
tervals\nlocal\n   Simp,  #current Simpson appprox\n   x,     #current
 x-value\n   dx,    #subinterval width\n   i;     #index\n\nSimp:=0: #
initialize\nx:=a:   #start at left endpoint\ndx:=evalf((b-a)/n);  #sub
interval length\nfor i from 0 to n-2 by 2 do\n  Simp:=Simp+f(a+i*dx)+4
*f(a+(i+1)*dx)+f(a+(i+2)*dx):\nod:\nSimp:=Simp*dx/3:\nreturn(Simp);\ne
nd:" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Let's test:" }}{PARA 0 "" 0 "" 
{TEXT -1 73 "Exercise 1)  By hand, work out the Trapezoid and Simpson \+
Approximation to" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(x^2,x = 0 .. \+
1);" "6#-%$IntG6$*$)%\"xG\"\"#\"\"\"/F(;\"\"!F*" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "with n=4 subdivisions" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Compare to " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#
\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Trapezoid(f
,0,1,4);\nSimpson(f,0,1,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++]
PM!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLLL!#5" }}}{PARA 0 "" 
0 "" {TEXT -1 106 "(In fact, Simpson is exact on any degree three or l
ower polynomial, see the error estimate from the text!)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Simpson(f,0,1,2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"+LLLLL!#5" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 
21 "Exercise 2:  Estimate" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Int(1/t,
t = 1 .. 2);" "6#-%$IntG6$*&\"\"\"F'%\"tG!\"\"/F(;F'\"\"#" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Using n=2, Trapezoid and Simpson. \+
 The exact value is the natural logarithm of 2, since" }}{PARA 260 "" 
0 "" {XPPEDIT 18 0 "ln(x) := Int(1/t,t = 1 .. x);" "6#>-%#lnG6#%\"xG-%
$IntG6$*&\"\"\"F,%\"tG!\"\"/F-;F,F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Compare h
and work to MAPLE:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "g:=x->
1/x:\nTrapezoid(g,1,2,2);\nSimpson(g,1,2,2);\nln(2.0);\nTrapezoid(g,1,
2,10);\nSimpson(g,1,2,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ILL$
3(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZWWWp!#5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+1=ZJp!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+ISrPp!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5B]Jp!#5" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Exercise 3:  Es
timate  " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "h:=x->s
qrt(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)o
peratorG%&arrowGF(-%%sqrtG6#,&\"\"\"F0*$)9$\"\"#F0!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "4*Simpson(h,0,1,10000); #Thi
s should approximate Pi\nevalf(Pi);  #Maple's approximation" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+x@fTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+aEfTJ!\"*" }}}}{MARK "42 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }