input courbes.mp;
 
 verbatimtex
 %&latex
 \documentclass{article}
 \begin{document}
 \def\vect#1{\vec #1}
 etex
 
 

 beginfig(1);
 path p[],q[],t[];
 repere(0,0,-1,3,-1,2,1cm,2cm);
 p1 = r_point(0,0)--r_point(0,1)--r_point(1,1)--r_point(1,0)--cycle;
 fill p1 withcolor 0.5*white;
 
 label.bot(btex $I$ etex,  r_point(1,0));
 label.rt(btex $K$ etex,  r_point(1,1));
 label.lft(btex $J$ etex,  r_point(0,1));
 label.bot(btex $\vect{i}$ etex,  r_point(0.5,0));
 label.lft(btex $\vect{j}$ etex,  r_point(0,0.5));
 label(btex $u.a.$ etex,  r_point(0.5,0.5));
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 %========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                cos(t)+4
 enddef;
 %========================================================================
 vardef gx(expr t) =
 t
 enddef;
 vardef gy(expr t) =
                cos(t-0.5)+3
 enddef;
 %======================================================================== 
 

 beginfig(2);
 path p[],q[],t[];
 repere(0,0,-1,9,-1,6,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 p2 = f_courbe(gx,gy,2,9,100);
 q1 = rx_droite(3);
 q2= rx_droite(8);
 q3 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q2,q3);
 fill t1 withcolor .8white;
 t2= buildcycle(q1,p2,q2,q3);
 unfill t2;
 fill t2 withcolor .4white;
 
 
 %trace de la surface
 draw p1;draw p2;
 draw r_point(3,0)--f_point(fx,fy,3);
 draw r_point(8,0)--f_point(fx,fy,8);
 label.lft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.lft(btex $y=g(x)$ etex, f_point(gx,gy,2));
 label.bot(btex $a$ etex,  r_point(3,0));
 label.bot(btex $b$ etex,  r_point(8,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 
 
 %========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                cos(t)+0.5
 enddef;
 %========================================================================
 vardef gx(expr t) =
 t
 enddef;
 vardef gy(expr t) =
                cos(t-0.5)-0.5
 enddef;
 %======================================================================== 
 

 beginfig(3);
 path p[],q[],t[];
 repere(0,0,-1,9,-3,3,1cm,1cm);
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 p2 = f_courbe(gx,gy,2,9,100);
 q1 = rx_droite(3);
 q2= rx_droite(7);
 q3 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(p2,q1,p1,q2);
 fill t1 withcolor .4white;
 
 %trace de la surface
 draw p1;draw p2;
 draw r_point(3,0)--f_point(gx,gy,3);
 draw r_point(7,0)--f_point(fx,fy,7);
 label.ulft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.lft(btex $y=g(x)$ etex, f_point(gx,gy,2));
 label.top(btex $a$ etex,  r_point(3,0));
 label.bot(btex $b$ etex,  r_point(7,0));
 r_fin;
 endfig;
 
 %========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                cos(t)+4
 enddef;
 %======================================================================== 
 

 beginfig(4);
 path p[],q[],t[];
 repere(0,0,-1,9,-1,6,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 q1 = rx_droite(3);
 q2= rx_droite(6);
 q3 =rx_droite(8);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q2,p1,q3,q4);
 fill t1 withcolor .4white;
 t2= buildcycle(q1,p1,q2,q4);
 fill t2 withcolor .9white;
 
 %trace de la surface
 draw p1;
 draw r_point(3,0)--f_point(fx,fy,3);
 draw r_point(6,0)--f_point(fx,fy,6);
 draw r_point(8,0)--f_point(fx,fy,8);
 label.lft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.bot(btex $a$ etex,  r_point(3,0));
 label.bot(btex $b$ etex,  r_point(6,0));
 label.bot(btex $c$ etex,  r_point(8,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 
 %=========================================================================
 

 beginfig(5);
 path p[],q[],t[];
 repere(0,0,-1,9,-1,8,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 q1 = rx_droite(3);
 q3 =rx_droite(8);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q3,q4);
 fill t1 withcolor .4white;
 
 %trace de la surface
 draw p1;
 draw r_point(3,0)--r_point(3,6);
 draw r_point(8,0)--r_point(8,6);
 draw r_point(0,6)--r_point(8,6);
 draw r_point(8,2)--r_point(0,2);
 label.lft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.bot(btex $a$ etex,  r_point(3,0));
 label.bot(btex $b$ etex,  r_point(8,0));
 
 label.ulft(btex $A$ etex,  r_point(3,0));
 label.urt(btex $B$ etex,  r_point(8,0));
 label.ulft(btex $F$ etex,  r_point(3,2));
 label.urt(btex $E$ etex,  r_point(8,2));
 label.top(btex $D$ etex,  r_point(3,6));
 label.top(btex $C$ etex,  r_point(8,6));
 label.lft(btex $M$ etex,  r_point(0,6));
 label.lft(btex $m$ etex,  r_point(0,2));
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 %======================================================================== 
 

 beginfig(6);
 path p[],q[],t[];
 repere(0,0,-1,9,-1,6,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 q1 = rx_droite(3);
 q2= rx_droite(6);
 q3 =rx_droite(8);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q3,q4);
 fill t1 withcolor .4white;
 
 %trace de la surface
 draw p1;
 draw r_point(3,0)--r_point(3,fy(7));
 %draw r_point(7,0)--f_point(fx,fy,7) dashed evenly;
 draw r_point(0,fy(7))--r_point(3,fy(7)) dashed evenly;
 draw r_point(8,0)--r_point(8,fy(7));
 draw r_point(3,fy(7))--r_point(8,fy(7));
 
 label.lft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.bot(btex $a$ etex,  r_point(3,0));
 %label.bot(btex $c$ etex,  r_point(7,0));
 label.bot(btex $b$ etex,  r_point(8,0));
 label.lft(btex $h$ etex,  r_point(0,fy(7)));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 

 beginfig(7);
 numeric u;
 u=1cm;
 path p[],q[],c[];
 %les axes et l'origine
 z0=(0u,0u);
 z1=(-3u,-3u);
 z2=(6u,0u);
 z3=(0u,10u);
 
 drawarrow z0--z1;
 drawarrow z0--z2;
 draw (0u,1u)--(0u,2u) dashed evenly ;
 draw (0u,4u)--(0u,5u) dashed evenly ;
 draw (0u,7u)--(0u,8u) dashed evenly ;
 draw (0u,0u)--(0u,1u);
 draw (0u,2u)--(0u,4u);
 draw (0u,5u)--(0u,7u);
 drawarrow (0u,8u)--(0u,10u);
 dotlabel.bot(btex $O$ etex, z0);
 label.lft(btex $x$ etex, z1);
 label.bot(btex $y$ etex, z2);
 label.lft(btex $z$ etex, z3);
 %les plans 
 draw (-2u,1u)--(-1u,3u)--(6u,3u)--(5u,1u)--cycle;
 draw (-2u,4u)--(-1u,6u)--(6u,6u)--(5u,4u)--cycle;
 draw (-2u,7u)--(-1u,9u)--(6u,9u)--(5u,7u)--cycle;
 
 label.lft(btex $a$ etex, (0u,2u));
 label.lft(btex $z$ etex, (0u,5u));
 label.lft(btex $b$ etex, (0u,8u));
 
 %le solide
 %le bas
 z10=(1u,2u);
 z11=(2u,2.8u);
 z12=(3u,2.5u);
 z13=(4u,2u);
 z14=(2u,1.3u);
  c1=z10{up}...{right}z11{right}...{right}z12{right}...{down}z13{down}...{left}z14{left}...cycle;
  fill c1 withcolor 0.5*white;
  draw c1;
 %le milieu
 z20=(1.5u,5u);
 z21=(2u,5.7u);
 z22=(3u,5.3u);
 z23=(3.5u,5u);
 z24=(2u,4.5u);
 c2=z20{up}...{right}z21{right}...{right}z22{right}...{down}z23{down}...{left}z24{left}...cycle;
 fill c2 withcolor 0.5*white;
 draw c2;
 %le haut
 z30=(1.5u,8u);
 z31=(2u,8.5u);
 z32=(3u,8.1u);
 z33=(3.5u,8u);
 z34=(2u,7.6u);
 c3=z30{up}...{right}z31{right}...{right}z32{right}...{down}z33{down}...{left}z34{left}...cycle;
 fill c3 withcolor 0.5*white;
 draw c3;
 
 %les generatrices
 q1 = z10...z20...z30;
 p1 = (-2u,4u)--(5u,4u);
 z40 = p1 intersectionpoint q1;
 draw z10...z40; draw z40...z20 dashed evenly;
 p2 = (-2u,7u)--(5u,7u);
 z41 = p2 intersectionpoint q1;
 draw z20...z41; draw z41...z30 dashed evenly;
 
 q2 = z13...z23...z33;
 z50 = p1 intersectionpoint q2;
 draw z13...z50; draw z50...z23 dashed evenly;
 z51 = p2 intersectionpoint q2;
 draw z23...z51; draw z51...z33 dashed evenly;
 
 q3 = z14...z24...z34;
 z60 = p1 intersectionpoint q3;
 draw z14...z60; draw z60...z24 dashed evenly;
 z61 = p2 intersectionpoint q3;
 draw z24...z61; draw z61...z34 dashed evenly;
 
 %surface
 label.rt(btex $S(z)$ etex, z23);
 
 endfig;
 
 %========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                cos(t)
 enddef;
 %========================================================================
 vardef gx(expr t) =
 t
 enddef;
 vardef gy(expr t) =
               sin(t)
 enddef;
 %======================================================================== 
 

 beginfig(8);
 path p[],q[],t[];
 repere(0,0,-1,3,-1,2,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,0,3.14/2,100);
 p2 = f_courbe(gx,gy,0,3.14/2,100);
 q1 = rx_droite(0);
 q2=rx_droite(3.14/2);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,p2);
 fill t1 withcolor .4white;
 t2= buildcycle(p1,p2,q2);
 fill t2 withcolor .4white;
 
 %trace de la surface
 draw p1;draw p2;
 label.bot(btex $\frac{\pi}{2}$ etex,  r_point(3.14/2,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 %========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                cos(t)+4
 enddef;
 %======================================================================== 
 

 beginfig(9);
 path p[],q[],t[];
 repere(0,0,-1,9,-1,6,1cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(fx,fy,2,9,100);
 q1 = rx_droite(3);
 q3 =rx_droite(8);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q3,q4);
 fill t1 withcolor .4white;
 %trace de la surface
 draw p1;
 draw r_point(3,0)--f_point(fx,fy,3);
 draw r_point(8,0)--f_point(fx,fy,8);
 label.lft(btex $y=f(x)$ etex, f_point(fx,fy,2));
 label.bot(btex $a$ etex,  r_point(3,0));
 label.bot(btex $b$ etex,  r_point(8,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 
 %======================================================================== 
 

 beginfig(10);
 pair x[],c[],p[];
 repere(0,0,-1,7,-2,6,1cm,1cm);
 %on place les points
 x0=r_point(2,0);
 x1=r_point(2.5,0);
 x2=r_point(3.5,0);
 x3=r_point(5,0);
 x4=r_point(6,0);
 
 c1=r_point(0,1);
 c2=r_point(0,4);
 c3=r_point(0,-1);
 c4=r_point(0,3);
 
 p0=r_point(2,1);
 p1=r_point(2.5,1);
 p2=r_point(2.5,4);
 p3=r_point(3.5,4);
 p4=r_point(3.5,-1);
 p5=r_point(5,-1);
 p6=r_point(5,3);
 p7=r_point(6,3);
 %trace des rectangles
 draw x0--p0 dashed evenly;
 draw p0--c1 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p0--p1 ;
 pickup pencircle scaled 0.5pt;
 draw x1--p2  dashed evenly;
 draw p2--c2 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p2--p3;
 pickup pencircle scaled 0.5pt;
 draw p3--p4  dashed evenly; 
 draw p4--c3 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p4--p5 ;
 pickup pencircle scaled 0.5pt;
 draw p5--p6 dashed evenly ;
 draw p6--c4 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p6--p7 ;
 pickup pencircle scaled 0.5pt;
  draw p7--x4 dashed evenly ;  
  
 
 %label sur les axes
 label.lft(btex $c_{1}$ etex,c1);
 label.lft(btex $c_{2}$ etex,c2);
 label.lft(btex $c_{3}$ etex,c3);
 label.lft(btex $c_{4}$ etex,c4);
 
 label.llft(btex $x_{0}=a$ etex,x0);
 label.bot(btex $x_{1}$ etex,x1);
 label.bot(btex $x_{2}$ etex,x2);
 label.bot(btex $x_{3}$ etex,x3);
 label.bot(btex $x_{4}=b$ etex,x4);
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 %======================================================================== 
 

 beginfig(11);
 pair x[],c[],p[];
 repere(0,0,-1,7,-2,6,1cm,1cm);
 %on place les points
 x0=r_point(2,0);
 x1=r_point(2.5,0);
 x2=r_point(3.5,0);
 x3=r_point(5,0);
 x4=r_point(6,0);
 
 c1=r_point(0,1);
 c2=r_point(0,4);
 c3=r_point(0,-1);
 c4=r_point(0,3);
 
 p0=r_point(2,1);
 p1=r_point(2.5,1);
 p2=r_point(2.5,4);
 p3=r_point(3.5,4);
 p4=r_point(3.5,-1);
 p5=r_point(5,-1);
 p6=r_point(5,3);
 p7=r_point(6,3);
 %trace des rectangles
 pickup pencircle scaled 1pt;
 draw x0--p0 ;
 pickup pencircle scaled 0.5pt;
 draw p0--c1 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p0--p1 ;
 draw x1--p2  ;
 pickup pencircle scaled 0.5pt;
 draw p2--c2 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p2--p3;
 draw p3--p4  ; 
 pickup pencircle scaled 0.5pt;
 draw p4--c3 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p4--p5 ;
 draw p5--p6  ;
 pickup pencircle scaled 0.5pt;
 draw p6--c4 dashed evenly ;
 pickup pencircle scaled 1pt;
 draw p6--p7 ;
  draw p7--x4  ;  
  pickup pencircle scaled 0.5pt;
  %marque + ou -
 label(btex $+$ etex, 0.5*(x0+p1));
 label(btex $+$ etex, 0.5*(x1+p3));
 label(btex $-$ etex, 0.5*(x2+p5));
 label(btex $+$ etex, 0.5*(x3+p7));
 
 %label sur les axes
 label.lft(btex $c_{1}$ etex,c1);
 label.lft(btex $c_{2}$ etex,c2);
 label.lft(btex $c_{3}$ etex,c3);
 label.lft(btex $c_{4}$ etex,c4);
 
 label.llft(btex $x_{0}=a$ etex,x0);
 label.bot(btex $x_{1}$ etex,x1);
 label.bot(btex $x_{2}$ etex,x2);
 label.bot(btex $x_{3}$ etex,x3);
 label.bot(btex $x_{4}=b$ etex,x4);
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 
 
 %==========================================================================
 vardef fx(expr t) =
 t
 enddef;
 vardef fy(expr t) =
                       t**2
 enddef;
 %==========================================================================
                 
 

 beginfig(12); %pour l'activite
 path p,q; 
 numeric h;
 repere(0,0,-0.2,1.2,-0.2,1.2,10cm,10cm);
 %quadunite_x(1,black);
 %quadunite_y(1,black);
 r_axes;
 r_origine;
 r_unites;
 r_labelxy;
 p =f_courbe(fx,fy,0,1,100);
 draw p;
 trace_rectangles_min(fx,fy,0,1,1/5);
 trace_rectangles_max(fx,fy,0,1,1/5);
 r_fin;
 endfig;
 
 

 beginfig(13);%pour l'activite
 path p,q; 
 numeric h;
 repere(0,0,-0.2,1.2,-0.2,1.2,10cm,10cm);
 %quadunite_x(1,black);
 %quadunite_y(1,black);
 r_axes;
 r_origine;
 r_unites;
 r_labelxy;
 p =f_courbe(fx,fy,0,1,100);
 draw p;
 trace_rectangles_min(fx,fy,0,1,1/10);
 trace_rectangles_max(fx,fy,0,1,1/10);
 r_fin;
 endfig;
 
 

 beginfig(14);%pour l'activite
 path p,q; 
 numeric h;
 repere(0,0,-0.2,1.2,-0.2,1.2,10cm,10cm);
 %quadunite_x(1,black);
 %quadunite_y(1,black);
 r_axes;
 r_origine;
 r_unites;
 r_labelxy;
 p =f_courbe(fx,fy,0,1,100);
 draw p;
 trace_rectangles_min(fx,fy,0,1,1/20);
 trace_rectangles_max(fx,fy,0,1,1/20);
 r_fin;
 endfig;
 
 %========================================================================
 vardef ix(expr t) =
 t
 enddef;
 vardef iy(expr t) =
                1/t
 enddef;
 %======================================================================== 
 

 beginfig(15);
 path p[],q[],t[];
 repere(0,0,-0.2,1.2,-1,10,10cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(ix,iy,0.01,1.2,100);
 q1 = rx_droite(0.4);
 q3 =rx_droite(1);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q3,q4);
 fill t1 withcolor .4white;
 %trace de la surface
 draw p1;
 draw r_point(0.4,0)--f_point(ix,iy,0.4);
 draw r_point(1,0)--f_point(ix,iy,1);
 label.urt(btex $y=\frac{1}{x}$ etex, f_point(ix,iy,0.7));
 label.urt(btex $\ln x {\textnormal{ si } 0<x<1}$ etex, r_point(0.4,3));
 label.bot(btex $x$ etex,  r_point(0.4,0));
 label.bot(btex $1$ etex,  r_point(1,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 
 endfig;
 
 %========================================================================
 vardef ix(expr t) =
 t
 enddef;
 vardef iy(expr t) =
                1/t
 enddef;
 %======================================================================== 
 

 beginfig(16);%pour ln
 path p[],q[],t[];
 repere(0,0,-0.2,2.2,-1,10,8cm,1cm);
 
 %definition de la surface
 p1 = f_courbe(ix,iy,0.01,2,100);
 q1 = rx_droite(1);
 q3 =rx_droite(1.8);
 q4 = r_droitedir(0,0,0);
 
 %remplissage de la surface
 t1 = buildcycle(q1,p1,q3,q4);
 fill t1 withcolor .4white;
 %trace de la surface
 draw p1;
 draw r_point(1,0)--f_point(ix,iy,1);
 draw r_point(1.8,0)--f_point(ix,iy,1.8);
 label.urt(btex $y=\frac{1}{x}$ etex, f_point(ix,iy,0.7));
 label.urt(btex $\ln x {\textnormal{ si } x>1}$ etex, r_point(1,3));
 label.bot(btex $1$ etex,  r_point(1,0));
 label.bot(btex $x$ etex,  r_point(1.8,0));
 
 %trace du repère
 r_axes;
 r_origine;
 %r_unites;
 %r_labelxy;
 r_fin;
 endfig;
 end