exo26.mw

> restart;

> with(plots);with(student);

Warning, the name changecoords has been redefined

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...

[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, completesquare, distance, equate, integrand, intercept, intparts, leftbox, leftsum, makeproc, middlebox, middlesum, midpoi...

Exercice 26

question a)

> Dom:={ x>=0,y>=0,x^2+y^2<=1,x<=x^2+y^2};

Dom := {x^2+y^2 <= 1, x <= x^2+y^2, 0 <= y, 0 <= x}

> f:=(x,y)-> x+y;

f := proc (x, y) options operator, arrow; x+y end proc

> implicitplot(Dom,x=0..1,y=0..1,thickness=3);

[Plot]

Changement de coordonnées polaires

> Doubleint('f'(x,y),x,y,A)= Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``)

domaine d'intégration polaire

> Dpol:={0<=theta,theta<=Pi/2,cos(theta)<=r,r<=1};

Dpol := {theta <= 1/2*Pi, r <= 1, cos(theta) <= r, 0 <= theta}

> implicitplot(Dpol,r=0..0.995,theta=0..1.55,thickness=3);

[Plot]

> Doubleint('f'(x,y),x,y,A)=  Doubleint('f'(r*cos(theta),r*sin(theta))*r,r=cos(theta)..1,theta=0..Pi/2) ,"=",Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=cos(theta)..1,theta=0..Pi/2) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*r,r=cos(theta)..1,theta=0..Pi/2) );

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*r, r = cos(theta) .. 1), theta = 0 .. 1/2*Pi),

question b)

> Dom:={  y>=0,x^2+y^2<=x,y<=x^2+y^2};

Dom := {0 <= y, x^2+y^2 <= x, y <= x^2+y^2}

> f:=(x,y)-> x^2+y^2;

f := proc (x, y) options operator, arrow; y^2+x^2 end proc

> implicitplot(Dom,x=0..1,y=0..0.5,thickness=3);

[Plot]

Changement de coordonnées polaires

> Doubleint('f'(x,y),x,y,A)= Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``)

domaine d'intégration polaire

> Dpol:={0<=theta,theta<=Pi/4,sin(theta)<=r,r<=cos(theta)};

Dpol := {theta <= 1/4*Pi, sin(theta) <= r, r <= cos(theta), 0 <= theta}

> implicitplot(Dpol,r=0..1,theta=0..0.75,thickness=3);

[Plot]

> Doubleint('f'(x,y),x,y,A)=  Doubleint('f'(r*cos(theta),r*sin(theta))*r,r=sin(theta)..cos(theta),theta=0..Pi/4) ,"=",Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=sin(theta)..cos(theta),theta=0..Pi/4) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*r,r=sin(theta)..cos(theta),theta=0..Pi/4) );

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*r, r = sin(theta) .. cos(theta)), theta = 0 .. 1/4*Pi),

>

question c)

> Dom:={  (x^2+y^2)^2<=2*x*y};

Dom := {(x^2+y^2)^2 <= 2*x*y}

> f:=(x,y)-> sqrt(x*y);

f := proc (x, y) options operator, arrow; sqrt(y*x) end proc

> implicitplot(Dom,x=-1..1,y=-1..1,thickness=3);

[Plot]

puisque le domaine d'intégration est symétrique par rapport à O et que

> 'f'(-x,-y)='f'(x,y);

f(-x, -y) = f(x, y)

On a

> Doubleint('f'(x,y),x,y,A)= 2 *Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,B);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 2*Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = B .. ``), theta = `` .. ``)

Où B est la partie de A dont les abscisses sont positives

> DomB:={  (x^2+y^2)^2<=2*x*y,x>=0};

DomB := {0 <= x, (x^2+y^2)^2 <= 2*x*y}

> implicitplot(Dom,x=0..1,y=-1..1,thickness=3);

[Plot]

Changement de coordonnées polaires

> Doubleint('f'(x,y),x,y,B)= Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Bpol);

Int(Int(f(x, y), x = B .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Bpol .. ``), theta = `` .. ``)

domaine d'intégration polaire

> Dpol:={0<=theta,theta<=Pi/2,r^2<=sin(2*theta)};

Dpol := {theta <= 1/2*Pi, r^2 <= sin(2*theta), 0 <= theta}

> implicitplot(Dpol,r=0..1,theta=-1.52..1.57,thickness=3);

[Plot]

>

>

> Doubleint('f'(x,y),x,y,B)=  Doubleint('f'(r*cos(theta),r*sin(theta))*r,r=0..sqrt(sin(2*theta)),theta=0..Pi/2) ,"=",Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..sqrt(sin(2*theta)),theta=0..Pi/2) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*r,r=0..sqrt(sin(2*theta)),theta=0..Pi/2) );

Int(Int(f(x, y), x = B .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*r, r = 0 .. sin(2*theta)^(1/2)), theta = 0 .. 1/2*Pi),

> Doubleint('f'(x,y),x,y,A)=2*1/24*2^(1/2)*Pi;

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 1/12*2^(1/2)*Pi

question d)

> Dom_polaire:={ abs(r)<=sqrt(cos(theta))};

Dom_polaire := {abs(r) <= cos(theta)^(1/2)}

> f:=(x,y)-> x^2;

f := proc (x, y) options operator, arrow; x^2 end proc

> implicitplot(Dom_polaire,r=-1..1,theta=-Pi/2..Pi/2,thickness=3);

[Plot]

Changement de coordonnées polaires

> Doubleint('f'(x,y),x,y,A)= Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``)

> Doubleint('f'(x,y),x,y,A)=  Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r=-sqrt(cos(theta))..sqrt(cos(theta)),theta=-Pi/2..Pi/2) ,"=",Doubleint(simplify(f(r*cos(theta),r*sin(theta))*abs(r)),r=-sqrt(cos(theta))..sqrt(cos(theta)),theta=-Pi/2..Pi/2) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*abs(r),r=-sqrt(cos(theta))..sqrt(cos(theta)),theta=-Pi/2..Pi/2) );

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = -cos(theta)^(1/2) .. cos(theta)^(1/2)), theta = -1/2*Pi .. 1/2*Pi),

>

question e)

> Dom:={ x>=0,y>=0,x+y<=1};

Dom := {0 <= x, x+y <= 1, 0 <= y}

> f:=(x,y)-> (x+y)^2*exp(x^2-y^2);;

f := proc (x, y) options operator, arrow; (x+y)^2*exp(x^2-y^2) end proc

> inequal(Dom,x=-1..2,y=-1..2,optionsexcluded=(color=white));

[Plot]

Changement de coordonnées affine

> u=x+y,v=x-y;

u = x+y, v = x-y

> Doubleint('f'((u+v)/2,(u-v)/2),u,v,B)= Doubleint('f'(x,y)*abs(2),x,y,A);

Int(Int(f(1/2*u+1/2*v, 1/2*u-1/2*v), u = B .. ``), v = `` .. ``) = Int(Int(2*f(x, y), x = A .. ``), y = `` .. ``)

Où le domaine d'intégration est le quart de cercle trigo C

> DomB:={ u+v>=0,u-v>=0,u<=2};

DomB := {0 <= u+v, 0 <= u-v, u <= 2}

> inequal(DomB,u=-1..2,v=-2..2,optionsexcluded=(color=white));

[Plot]

> Doubleint('f'(x,y),x,y,A)=1/2*Doubleint('f'((u+v)/2,(u-v)/2),v=-u..u,u=0..1),"=",1/2*Doubleint(simplify(f((u+v)/2,(u-v)/2)),v=-u..u,u=0..1),"=",value(1/2*Doubleint(f((u+v)/2,(u-v)/2),v=-u..u,u=0..1)) ;

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 1/2*Int(Int(f(1/2*u+1/2*v, 1/2*u-1/2*v), v = -u .. u), u = 0 .. 1),

question f)

> Dom:={ x*y>=1,y>=x,x+y>=0,x+y<=4};

Dom := {x+y <= 4, 0 <= x+y, 1 <= x*y, x <= y}

> f:=(x,y)-> (x^2-y^2)*cos(x*y);;

f := proc (x, y) options operator, arrow; (x^2-y^2)*cos(y*x) end proc

> implicitplot(Dom,x=0..4,y=-1..4,thickness=3);

[Plot]

Changement de coordonnées affine

> u=x+y,v=x-y;

u = x+y, v = x-y

> Doubleint('f'((u+v)/2,(u-v)/2),u,v,B)= Doubleint('f'(x,y)*abs(2),x,y,A);

Int(Int(f(1/2*u+1/2*v, 1/2*u-1/2*v), u = B .. ``), v = `` .. ``) = Int(Int(2*f(x, y), x = A .. ``), y = `` .. ``)

Où le domaine d'intégration est le quart de cercle trigo C

> DomB:={ (u+v)*(u-v)>=1,u-v>=u+v,u<=4,u>=0};

DomB := {u <= 4, 1 <= (u+v)*(u-v), u+v <= u-v, 0 <= u}

> implicitplot(DomB,u=0..4,v=-4..0,thickness=3);

[Plot]

> Doubleint('f'(x,y),x,y,A)=1/2*Doubleint('f'((u+v)/2,(u-v)/2),v=-sqrt(u^2-4)..0,u=2..4),"=",1/2*Doubleint(simplify(f((u+v)/2,(u-v)/2)),v=-sqrt(u^2-4)..0,u=2..4),"=",value(1/2*Doubleint(f((u+v)/2,(u-v)/2),v=-sqrt(u^2-4)..0,u=2..4)) ;

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 1/2*Int(Int(f(1/2*u+1/2*v, 1/2*u-1/2*v), v = -(u^2-4)^(1/2) .. 0), u = 2 .. 4),

>