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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 259 21 "Int\351gration num\351r
ique" }}{PARA 0 "" 0 "" {TEXT 260 13 "2 Motivations" }{TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f:=x->exp(-x^2);\nint(f(x),
x=-1..1);\nI1:=evalf(int(f(x),x=-1..1));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"
\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$erfG6#\"\"\"F'
%#PiG#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G$\"+m#[O\\\"!\"
*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 157 "Pour calculer cette int
\351grale, Maple utilise la fonction d'erreur. Au vu de la d\351finiti
on de cette fonction, la r\351ponse propos\351e n'est pas du tout expl
icite." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 257 "" 0 "" {TEXT 256 27 "4 La m\351thode des rectangles
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 40 "4.1 L
a m\351thode des rectangles \340 gauche  " }{TEXT -1 3 "\n1." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h=(b-a)/n;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "2. Sg(f) devient : " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "Sg(g)=h*sum(g(a[i]),i=0..n-1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%#SgG6#%\"gG*&%\"hG\"\"\"-%$sumG6$-F'6#&%\"aG6
#%\"iG/F3;\"\"!,&%\"nGF*F*!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
2 "3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a[i]=a+i*h;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"iG,&F%\"\"\"*&F'F)%\"hGF)F
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "4." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 105 "rectangleg1:=proc(f,a,b,n)\n             local h;
\nh:=evalf((b-a)/n);\nh*add(evalf(f(a+i*h)),i=0..n-1);\nend;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%,rectangleg1Gf*6&%\"fG%\"aG%\"bG%\"nG6#%\"
hG6\"F-C$>8$-%&evalfG6#*&,&9&\"\"\"9%!\"\"F79'F9*&F0F7-%$addG6$-F26#-9
$6#,&F8F7*&%\"iGF7F0F7F7/FF;\"\"!,&F:F7F7F9F7F-F-F-" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 2 "5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"rectangleg1(f,-1,1,1000);\nI1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+vxk$\\\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "6." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 178 "rectangleg2:=proc(f,a,b,n)\n             local M1,I1
;\nM1:=max(seq(evalf(abs(D(f)(a+k*(b-a)/n))),k=0..n));\nI1:=rectangleg
1(f,a,b,n);\n[I1-M1*(b-a)^2/(2*n),I1+M1*(b-a)^2/(2*n)];\nend;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,rectangleg2Gf*6&%\"fG%\"aG%\"bG%\"n
G6$%#M1G%#I1G6\"F.C%>8$-%$maxG6#-%$seqG6$-%&evalfG6#-%$absG6#--%\"DG6#
9$6#,&9%\"\"\"*(%\"kGFF,&9&FFFE!\"\"FF9'FKFF/FH;\"\"!FL>8%-%,rectangle
g1G6&FBFEFJFL7$,&FQFF*&#FF\"\"#FF*(F1FFFIFYFLFKFFFK,&FQFF*&#FFFYFFFZFF
FFF.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "7." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "rectangleg2(f,-1,1,1000);\nI1;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7$$\"+]A$>\\\"!\"*$\"++LO&\\\"F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "Remarque : On observe que I1 est bien dans l'intervalle p
ropos\351." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 261 0 "" }{TEXT 262 39 "4.2 La m\351thode des rectangle
s \340 droite " }{TEXT -1 3 "\n1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "rectangled1:=proc(f,a,b,n)\n             local h;\nh
:=evalf((b-a)/n);\nh*add(evalf(f(a+i*h)),i=1..n);\nend;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%,rectangled1Gf*6&%\"fG%\"aG%\"bG%\"nG6#%\"hG6
\"F-C$>8$-%&evalfG6#*&,&9&\"\"\"9%!\"\"F79'F9*&F0F7-%$addG6$-F26#-9$6#
,&F8F7*&%\"iGF7F0F7F7/FF;F7F:F7F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "rectangled1(f,-1,1,1000);\nI1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+vxk$\\\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+m#[O\\\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Nous retrouvons
 le m\352me r\351sultat qu'avec la m\351thode des rectangles \340 gauc
he car la fonction f est paire." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 
"2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "rectangled2:=proc(f
,a,b,n)\n             local M1,I1;\nM1:=max(seq(evalf(abs(D(f)(a+k*(b-
a)/n))),k=0..n));\nI1:=rectangleg1(f,a,b,n);\n[I1-M1*(b-a)^2/(2*n),I1+
M1*(b-a)^2/(2*n)];\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,rectang
led2Gf*6&%\"fG%\"aG%\"bG%\"nG6$%#M1G%#I1G6\"F.C%>8$-%$maxG6#-%$seqG6$-
%&evalfG6#-%$absG6#--%\"DG6#9$6#,&9%\"\"\"*(%\"kGFF,&9&FFFE!\"\"FF9'FK
FF/FH;\"\"!FL>8%-%,rectangleg1G6&FBFEFJFL7$,&FQFF*&#FF\"\"#FF*(F1FFFIF
YFLFKFFFK,&FQFF*&#FFFYFFFZFFFFF.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 2 "3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rectangled2(f,-
1,1,1000);\nI1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+]A$>\\\"!\"*$
\"++LO&\\\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Remarque : On observe que I1 est b
ien dans l'intervalle propos\351." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 263 25 "5 La m\351thode des trap\350zes" }{TEXT 
-1 3 "\n1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "trapeze1:=pro
c(f,a,b,n);\n(rectangleg1(f,a,b,n)+rectangled1(f,a,b,n))/2;\nend;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)trapeze1Gf*6&%\"fG%\"aG%\"bG%\"nG6
\"F+F+,&*&#\"\"\"\"\"#F/-%,rectangleg1G6&9$9%9&9'F/F/*&F.F/-%,rectangl
ed1GF3F/F/F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "2." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "trapeze1(f,-1,1,1000);\nI1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vxk$\\\"!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
84 "Nous retrouvons la m\352me approximation de I1 pour les m\352mes r
aisons que pr\351c\351demment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 2 "3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 176 "trapeze2:=proc(f,a,b,n)\n          local M2,I1;\nM2:=max(seq(
evalf(abs(D[1$2](f)(a+i*(b-a)/n))),i=0..n));\nI1:=trapeze1(f,a,b,n);\n
[I1-M2*(b-a)^3/12/n^2,I1+M2*(b-a)^3/12/n^2];\nend;" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%)trapeze2Gf*6&%\"fG%\"aG%\"bG%\"nG6$%#M2G%#I1G6\"F.
C%>8$-%$maxG6#-%$seqG6$-%&evalfG6#-%$absG6#--&%\"DG6#-%\"$G6$\"\"\"\"
\"#6#9$6#,&9%FF*(%\"iGFF,&9&FFFL!\"\"FF9'FQFF/FN;\"\"!FR>8%-%)trapeze1
G6&FIFLFPFR7$,&FWFF*&#FF\"#7FF*(F1FFFO\"\"$FR!\"#FFFQ,&FWFF*&#FFFinFFF
jnFFFFF.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "trapeze2(f,
-1,1,1000);\nI1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+Ukk$\\\"!\"*
$\"+3\"\\O\\\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Remarque : Le r\351sultat obte
nu est le m\352me dans les trois m\351thodes (car f est paire). Cepend
ant, nous obtenonsun intervalle moins large par cette m\351thode." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 23 "4 La m
\351thode de Simpson" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "1.
\nremarquons : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "(a[i]+a[
i+1])/2=collect(simplify((a+i*h+a+(i+1)*h)/2),h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&#\"\"\"\"\"#F'&%\"aG6#%\"iGF'F'*&F&F'&F*6#,&F,F'F'
F'F'F',&*&,&F,F'F&F'F'%\"hGF'F'F*F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 175 "simpson1:=proc(f,a,b,n)\n          local h;\nh:=eval
f((b-a)/n);\nh/6*( evalf(f(a)) + 4*add(evalf(f(a+(i+1/2)*h)),i=0..n-1)
 + 2*add(evalf(f(a+i*h)),i=1..n-1) + evalf(f(b)) );\nend;" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%)simpson1Gf*6&%\"fG%\"aG%\"bG%\"nG6#%\"hG6\"F
-C$>8$-%&evalfG6#*&,&9&\"\"\"9%!\"\"F79'F9,$*&#F7\"\"'F7*&F0F7,*-F26#-
9$6#F8F7*&\"\"%F7-%$addG6$-F26#-FD6#,&F8F7*&,&%\"iGF7#F7\"\"#F7F7F0F7F
7/FR;\"\"!,&F:F7F7F9F7F7*&FTF7-FI6$-F26#-FD6#,&F8F7*&FRF7F0F7F7/FR;F7F
XF7F7-F26#-FD6#F6F7F7F7F7F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 
"2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simpson1(f,-1,1,1000
);\nI1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l#[O\\\"!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 94 "On remarque que le r\351sultat propos\351 est bien plus p
r\351cis que celui des proc\351dures pr\351c\351dentes." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 186 "simpson2:=proc(f,a,b,n)\n          local M4,I1;\nM4:=max(seq(
evalf(abs(D[1$4](f)(a+i*(b-a)/n))),i=0..n));\nI1:=simpson1(f,a,b,n);\n
[I1-M4*(b-a)^5/(2880*n^4),I1+M4*(b-a)^5/(2880*n^4)];\nend;\n\n" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)simpson2Gf*6&%\"fG%\"aG%\"bG%\"nG6$
%#M4G%#I1G6\"F.C%>8$-%$maxG6#-%$seqG6$-%&evalfG6#-%$absG6#--&%\"DG6#-%
\"$G6$\"\"\"\"\"%6#9$6#,&9%FF*(%\"iGFF,&9&FFFL!\"\"FF9'FQFF/FN;\"\"!FR
>8%-%)simpson1G6&FIFLFPFR7$,&FWFF*&#FF\"%!)GFF*(F1FFFO\"\"&FR!\"%FFFQ,
&FWFF*&#FFFinFFFjnFFFFF.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "4.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simpson2(f,-1,1,1000);
\nI1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+l#[O\\\"!\"*F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+m#[O\\\"!\"*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 101 "La pr\351cision des calculs n'est pas suffisante.
 Nous reprenons les calculs en augmentant la pr\351cision." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Digits:=15;\nsimpson2(f,-1,1,1000);
\nevalf(int(f(x),x=-1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Digi
tsG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"0uCcE[O\\\"!#9$\"0+DcE
[O\\\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0'[il#[O\\\"!#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 52 }
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