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{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->exp(-x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)
9$\"\"#\"\"\"!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"TV:=int(f(x),x=1..1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TVG$\"+
3Ek$4\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Closed Newton-Cotes
" }}{PARA 0 "" 0 "" {TEXT -1 17 "Trapezoid (n = 1)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 41 "(0.5/2)*(f(1)+f(1.5));evalf(%);abs(%-TV);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+++++D!#5\"\"\"-%$expG6#!\"\"
F(F($\"+:1)\\j#!#6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+km>$=\"!#5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")cSb*)!#5" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 15 "Simpson (n = 2)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "(0.5/6)*(f(1)+4*f(1.25)+f(1.5));evalf(%);abs(%-TV);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+MLLL$)!#6\"\"\"-%$expG6#!\"
\"F(F($\"+8JPlyF'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7N5$4\"!#5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"''4R&!#5" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 19 "Simpson 3/8 (n = 3)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "(0.5/8)*(f(1)+3*f(7/6)+3*f(4/3)+f(1.5));evalf(%);abs(
%-TV);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&$\"++++]i!#6\"\"\"-%$exp
G6#!\"\"F(F(*&$\"++++v=!#5F(-F*6##!#\\\"#OF(F(*&F.F(-F*6##!#;\"\"*F(F(
$\"+Q:X(e'!#7F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wOS$4\"!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'K*Q#!#5" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Open Newto
n-Cotes" }}{PARA 0 "" 0 "" {TEXT -1 16 "Midpoint (n = 0)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "(1/2)*f(1.25); evalf(%);abs(%-TV);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Op0[5!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+Op0[5!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")sce
X!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "n=1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "(1/4)*(f(7/6)+f(4/3));evalf(%);abs(%-TV);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"%F&-%$expG6##!#\\\"#OF&
F&*&F%F&-F)6##!#;\"\"*F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!osM
1\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")G*p,$!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "n=2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 57 "(1/6)*(2*f(1.125)-f(1.25)+2*f(1.375));evalf(%);abs(%-TV);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*e:T4\"!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+*e:T4\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'
\")HZ!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Gaussian Quadrature" }}{PARA 0 "" 0 "" 
{TEXT -1 38 "Transformation (x = ((b-a)t + a + b)/2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "xt:=r->(r+5)/4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#xtGf*6#%\"rG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"
\"%F/9$F/F/#\"\"&F0F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "n =
 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r2:=fsolve(LegendreP(
2,x)=0,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G$\"+#p-Nx&!#5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "(1/4)*(f(xt(-r2))+f(xt(
r2)));abs(%-TV);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7E+%4\"!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'/+O!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 5 "n = 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r3:=fs
olve(LegendreP(3,x)=0,x=0.01..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#r3G$\"+#pmfu(!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c31:=
5/9.;c30:=8/9.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c31G$\"+cbbbb!#5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c30G$\"+*)))))))))!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "(1/4)*(c31*f(xt(-r3))+c30*f(
xt(0))+c31*f(xt(r3))); \nevalf(%);abs(%-TV);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&$\"+!y)Qyi!#6\"\"\"*&$\"+AAAAA!#5F'-%$expG6##!#D\"#;F
'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h>k$4\"!#5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"$Z'!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "
The c_i's are found from the formula in Theorem 4.7." }}{PARA 0 "" 0 "
" {TEXT -1 11 "For example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "c31:=int((x*(x-r3)/((-r3)*(-2*r3))),x=-1..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$c31G$\"+dbbbb!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 5 "n = 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "r41:=fsolve(
LegendreP(4,x)=0,x=0..0.5);\nr42:=fsolve(LegendreP(4,x)=0,x=0.5..1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r41G$\"+O/\")*R$!#5" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$r42G$\"+;JO6')!#5" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 159 "c41:=int(((x-r41)*(x-r42)*(x+r42)/((-r41-r41)*(-r4
1-r42)*(-r41+r42))),x=-1..1);\nc42:=int(((x-r41)*(x-r42)*(x+r41)/((-r4
2-r41)*(-r42-r42)*(-r42+r41))),x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$c41G$\"+\\:X@l!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$c42G$
\"+^%[&yM!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "(1/4)*(c42*
f(xt(-r42))+c41*f(xt(-r41))+c41*f(xt(r41))+c42*f(xt(r42))); \nevalf(%)
;abs(%-TV);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0Ek$4\"!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0Ek$4\"!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"$!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "32 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }