// xcas version=0.6.0 fontsize=16 // fltk 7Fl_Tile 33 -3240 1627 316 16 [ // fltk N4xcas7EditeurE 33 -3240 1627 245 16 455 , Rect(f,N,a,b):={ local RV,RR,R,k,av,ap,C; RV:=NULL; RR:=NULL; for(k:=0;k<=N;k++){ av:=a+k*(b-a)/N;//avant... ap:=a+(k+1)*(b-a)/N;//après... R:=polygone(point(av,0),point(av,f(av)),point(ap,f(av)),point(ap,0));//le rectangle RR:=RR,couleur(R,cyan+rempli);//on remplit les rectangles RV:=RV,couleur(R,bleu+line_width_3);//le contour en plus épais } C:=plot(f(x),x=a..b,affichage=rouge+line_width_3);//la courbe return(RV,RR,C);//les 3 graphes superposés }:;, // fltk N4xcas10Log_OutputE 33 -2995 1627 36 16 // Parsing Rect£// Warning: x declared as global variable(s) compiling Rect£ , // fltk N4xcas8EquationE 33 -2959 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 -2922 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -2922 1627 26 16 On choisit un élément n qui nous donnera le nombre de rectangles. , // fltk N4xcas10Log_OutputE 33 -2896 1627 1 16 ] , // fltk 7Fl_Tile 33 -2893 1627 743 16 [ // fltk N4xcas6FigureE 33 -2893 1627 742 16 // fltk N4xcas12History_PackE 35 -2871 537 133 16 [ // fltk 7Fl_Tile 46 -2871 526 50 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -2871 526 26 16 n:=element(1 .. 10,9.99) , // fltk N4xcas10Log_OutputE 46 -2845 526 1 16 , // fltk N4xcas10Gen_OutputE 46 -2844 526 23 16 parameter(n,1,10,9.99) ] , // fltk 7Fl_Tile 46 -2819 526 50 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -2819 526 26 16 Rect(x->cos(x),n,0,2*Pi) , // fltk N4xcas10Log_OutputE 46 -2793 526 1 16 , // fltk N4xcas10Gen_OutputE 46 -2792 526 23 16 Too large for display ] , // fltk 7Fl_Tile 46 -2767 526 27 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -2767 526 26 16 , // fltk N4xcas10Log_OutputE 46 -2741 526 1 16 // Parsing Trap£// Warning: x declared as global variable(s) compiling Trap£ ] ] // fltk N4xcas5Geo2dE 576 -2871 677 720 16 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, // fltk N4xcas10Log_OutputE 33 -2151 1627 1 16 ] , // fltk 7Fl_Tile 33 -2148 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -2148 1627 26 16 On fait la même chose en modifiant juste les coordonnées d'un point du polygone. , // fltk N4xcas10Log_OutputE 33 -2122 1627 1 16 ] , // fltk 7Fl_Tile 33 -2119 1627 367 16 [ // fltk N4xcas7EditeurE 33 -2119 1627 296 16 343 , Trap(f,N,a,b):={ local RV,RR,R,k,av,ap,C; RV:=NULL; RR:=NULL; for(k:=0;k<=N;k++){ av:=a+k*(b-a)/N; ap:=a+(k+1)*(b-a)/N; R:=polygone(point(av,0),point(av,f(av)),point(ap,f(ap)),point(ap,0));//le trapèze RR:=RR,couleur(R,cyan+rempli); RV:=RV,couleur(R,bleu+line_width_3); } C:=plot(f(x),x=a..b,affichage=rouge+line_width_3); return(RV,RR,C); }:;, // fltk N4xcas10Log_OutputE 33 -1823 1627 36 16 // Parsing Trap£// Warning: x declared as global variable(s) compiling Trap£ , // fltk N4xcas8EquationE 33 -1787 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 -1750 1627 27 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -1750 1627 26 16 , // fltk N4xcas10Log_OutputE 33 -1724 1627 1 16 ] , // fltk 7Fl_Tile 33 -1721 1627 475 16 [ // fltk N4xcas6FigureE 33 -1721 1627 474 16 // fltk N4xcas12History_PackE 35 -1699 343 162 16 [ // fltk 7Fl_Tile 46 -1699 332 27 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -1699 332 26 16 , // fltk N4xcas10Log_OutputE 46 -1673 332 1 16 ] , // fltk 7Fl_Tile 46 -1670 332 50 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -1670 332 26 16 n:=element(1 .. 10,5.04) , // fltk N4xcas10Log_OutputE 46 -1644 332 1 16 , // fltk N4xcas10Gen_OutputE 46 -1643 332 23 16 parameter(n,1,10,5.04) ] , // fltk 7Fl_Tile 46 -1618 332 50 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -1618 332 26 16 Trap(x->cos(x),floor(n),0,2*Pi) , // fltk N4xcas10Log_OutputE 46 -1592 332 1 16 , // fltk N4xcas10Gen_OutputE 46 -1591 332 23 16 Too large for display ] , // fltk 7Fl_Tile 46 -1566 332 27 16 [ // fltk N4xcas19Multiline_Input_tabE 46 -1566 332 26 16 , // fltk N4xcas10Log_OutputE 46 -1540 332 1 16 // Parsing aireRect£// Success compiling aireRect£ ] ] // fltk N4xcas5Geo2dE 382 -1699 865 452 16 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, // fltk N4xcas10Log_OutputE 33 -1247 1627 1 16 ] , // fltk 7Fl_Tile 33 -1244 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -1244 1627 26 16 On passe à l'approximation numérique en utilisant sum. , // fltk N4xcas10Log_OutputE 33 -1218 1627 1 16 ] , // fltk 7Fl_Tile 33 -1215 1627 172 16 [ // fltk N4xcas7EditeurE 33 -1215 1627 101 16 76 , aireRect(f,N,a,b):={ local k; evalf(sum((b-a)/N*f(a+k*(b-a)/N),k=1..N)); }:;, // fltk N4xcas10Log_OutputE 33 -1114 1627 36 16 // Parsing aireRect£// Success compiling aireRect£ , // fltk N4xcas8EquationE 33 -1078 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 -1041 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -1041 1627 26 16 aireRect(x->x^2,10,0,1) , // fltk N4xcas10Log_OutputE 33 -1015 1627 1 16 , // fltk N4xcas8EquationE 33 -1014 1627 35 16 0.385 ] , // fltk 7Fl_Tile 33 -977 1627 27 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -977 1627 26 16 , // fltk N4xcas10Log_OutputE 33 -951 1627 1 16 ] , // fltk 7Fl_Tile 33 -948 1627 172 16 [ // fltk N4xcas7EditeurE 33 -948 1627 101 16 94 , aireTrap(f,N,a,b):={ local k; evalf((b-a)/N*((f(a)+f(b))/2+sum(f(a+k*(b-a)/N),k=1..N-1))); }:;, // fltk N4xcas10Log_OutputE 33 -847 1627 36 16 // Parsing aireTrap£// Success compiling aireTrap£ , // fltk N4xcas8EquationE 33 -811 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 -774 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -774 1627 26 16 aireTrap(x->x^2,10,0,1) , // fltk N4xcas10Log_OutputE 33 -748 1627 1 16 , // fltk N4xcas8EquationE 33 -747 1627 35 16 0.335 ] , // fltk 7Fl_Tile 33 -710 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -710 1627 26 16 Alors que le résultat attendu est , // fltk N4xcas10Log_OutputE 33 -684 1627 1 16 ] , // fltk 7Fl_Tile 33 -681 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -681 1627 26 16 evalf(int(x^2,x=0..1)) , // fltk N4xcas10Log_OutputE 33 -655 1627 1 16 , // fltk N4xcas8EquationE 33 -654 1627 35 16 0.333333333333 ] , // fltk 7Fl_Tile 33 -617 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -617 1627 26 16 Pour la somme de Riemann, on utilise sum_riemann , // fltk N4xcas10Log_OutputE 33 -591 1627 1 16 ] , // fltk 7Fl_Tile 33 -588 1627 173 16 [ // fltk N4xcas7EditeurE 33 -588 1627 102 16 61 , Riem(f,a,b):={ sum_riemann(((b-a)/P)*f(a+p*(b-a)/P),[P,p])}:;, // fltk N4xcas10Log_OutputE 33 -486 1627 36 16 // Parsing Riem£// Warning: P p declared as global variable(s) compiling Riem£ , // fltk N4xcas8EquationE 33 -450 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 -413 1627 82 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -413 1627 26 16 Riem(x->x^2,-5,2) , // fltk N4xcas10Log_OutputE 33 -387 1627 1 16 , // fltk N4xcas8EquationE 33 -386 1627 55 16 133/3 ] , // fltk 7Fl_Tile 33 -329 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -329 1627 26 16 On vérifie , // fltk N4xcas10Log_OutputE 33 -303 1627 1 16 ] , // fltk 7Fl_Tile 33 -300 1627 82 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -300 1627 26 16 int(x^2,x=-5..2) , // fltk N4xcas10Log_OutputE 33 -274 1627 1 16 , // fltk N4xcas8EquationE 33 -273 1627 55 16 133/3 ] , // fltk 7Fl_Tile 33 -216 1627 26 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -216 1627 25 16 CALCULS DE PI , // fltk N4xcas10Log_OutputE 33 -191 1627 1 16 ] , // fltk 7Fl_Tile 33 -188 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -188 1627 26 16 Avec la méthode des rectangles appliqué à x->sqrt(1-x^2) on obtient , // fltk N4xcas10Log_OutputE 33 -162 1627 1 16 ] , // fltk 7Fl_Tile 33 -159 1627 81 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -159 1627 26 16 4*aireRect(x->sqrt(1-x^2),10000,0,1) , // fltk N4xcas10Log_OutputE 33 -133 1627 20 16 Evaluation time: 0.45£ , // fltk N4xcas8EquationE 33 -113 1627 35 16 3.14139147761 ] , // fltk 7Fl_Tile 33 -76 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 -76 1627 26 16 Avec la méthode des trapèzes appliqué à x->sqrt(1-x^2) on obtient , // fltk N4xcas10Log_OutputE 33 -50 1627 1 16 ] , // fltk 7Fl_Tile 33 -47 1627 81 16 [ // fltk N4xcas19Multiline_Input_tabE 33 -47 1627 26 16 4*aireTrap(x->sqrt(1-x^2),10000,0,1) , // fltk N4xcas10Log_OutputE 33 -21 1627 20 16 Evaluation time: 0.31£ , // fltk N4xcas8EquationE 33 -1 1627 35 16 3.14159147761 ] , // fltk 7Fl_Tile 33 36 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 36 1627 26 16 Alors qu'on devrait obtenir , // fltk N4xcas10Log_OutputE 33 62 1627 1 16 ] , // fltk 7Fl_Tile 33 65 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 65 1627 26 16 evalf(Pi) , // fltk N4xcas10Log_OutputE 33 91 1627 1 16 , // fltk N4xcas8EquationE 33 92 1627 35 16 3.14159265359 ] , // fltk 7Fl_Tile 33 129 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 129 1627 26 16 On note quand même que la méthode des trapèzes est plus efficace. , // fltk N4xcas10Log_OutputE 33 155 1627 1 16 ] , // fltk 7Fl_Tile 33 158 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 158 1627 26 16 FORMULE DE MACHIN , // fltk N4xcas10Log_OutputE 33 184 1627 1 16 ] , // fltk 7Fl_Tile 33 187 1627 201 16 [ // fltk N4xcas7EditeurE 33 187 1627 139 16 82 , mini(u,p):={ local k; k:=0; while(evalf((u^(2*k+3))/(2*k+3))>10^(-p)){ k:=k+1} }:;, // fltk N4xcas10Log_OutputE 33 326 1627 36 16 // Parsing mini£// Success compiling mini£ , // fltk N4xcas8EquationE 33 362 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 390 1627 27 16 [ // fltk N4xcas19Multiline_Input_tabE 33 390 1627 26 16 , // fltk N4xcas10Log_OutputE 33 416 1627 1 16 ] , // fltk 7Fl_Tile 33 419 1627 253 16 [ // fltk N4xcas7EditeurE 33 419 1627 182 16 116 , greg(n,a,p):={ local S,k; S:=0; for(k:=0;k<=n+1;k++){ S:=S+evalf(((-1)^k*a^(2*k+1))/(2*k+1),p+1); }; return(S); }:; , // fltk N4xcas10Log_OutputE 33 601 1627 36 16 // Parsing greg£// Success compiling greg£ , // fltk N4xcas8EquationE 33 637 1627 35 16 "Done" ] , // fltk 7Fl_Tile 33 674 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 674 1627 26 16 mini(1/5,100) , // fltk N4xcas10Log_OutputE 33 700 1627 1 16 , // fltk N4xcas8EquationE 33 701 1627 35 16 69 ] , // fltk 7Fl_Tile 33 738 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 738 1627 26 16 Il faut aller jusqu'au rang 69 pour avoir 100 bonnes décimales , // fltk N4xcas10Log_OutputE 33 764 1627 1 16 ] , // fltk 7Fl_Tile 33 767 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 767 1627 26 16 4*(4*greg(69,1/5,100)-greg(69,1/239,100)) , // fltk N4xcas10Log_OutputE 33 793 1627 1 16 , // fltk N4xcas8EquationE 33 794 1627 35 16 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706804 ] , // fltk 7Fl_Tile 33 831 1627 62 16 [ // fltk N4xcas19Multiline_Input_tabE 33 831 1627 26 16 4*(4*greg(69,1/5,100)-greg(69,1/239,100))-evalf(Pi,100) , // fltk N4xcas10Log_OutputE 33 857 1627 1 16 , // fltk N4xcas8EquationE 33 858 1627 35 16 -0.28574684782056874555394588707633523986297749870569966530378146080690815897668454891886200546504300741e-100 ] , // fltk 7Fl_Tile 33 895 1627 27 16 [ // fltk N4xcas23Comment_Multiline_InputE 33 895 1627 26 16 Pas mal... , // fltk N4xcas10Log_OutputE 33 921 1627 1 16 ] , // fltk 7Fl_Tile 33 924 1627 27 16 [ // fltk N4xcas19Multiline_Input_tabE 33 924 1627 26 16 , // fltk N4xcas10Log_OutputE 33 950 1627 1 16 ]