exo24.mw

> restart;

> with(plots);with(student);

Warning, the name changecoords has been redefined

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

Exercice 24

>

question a)

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

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

> f:=(x,y)->y/(x^2+1);

f := proc (x, y) options operator, arrow; y/(x^2+1) end proc

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

[Plot]

> Doubleint(f(x,y),x,y,A)= 0;

Int(Int(y/(1+x^2), x = A .. ``), y = `` .. ``) = 0

En effet, A est symétrique par rapport à (Ox) et

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

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

Plus particulièrement en notant Ap et Am les parties de A d'ordonnée positve resp négative on a

> Doubleint('f'(x,y),x,y,A)=Doubleint('f'(x,y),x,y,Ap)+Doubleint('f'(x,y),x,y,Am);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(x, y), x = Ap .. ``), y = `` .. ``)+Int(Int(f(x, y), x = Am .. ``), y = `` .. ``)

> Or par changement de variable u=x,v=-y;

u = x, v = -y

> Doubleint('f'(u,v),u,v,Am)=Doubleint('f'(x,-y),x,y,Ap),"=",-Doubleint('f'(x,y),x,y,Ap);

Int(Int(f(u, v), u = Am .. ``), v = `` .. ``) = Int(Int(f(x, -y), x = Ap .. ``), y = `` .. ``),

>

question b)

> Dom:={x^4+y^4+x^2-y^2<=1};

Dom := {x^4+y^4+x^2-y^2 <= 1}

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

f := proc (x, y) options operator, arrow; y/(x^2+1) end proc

> implicitplot(Dom,x=-1..1,y=-2..2,thickness=3);

[Plot]

> Doubleint(f(x,y),x,y,A)= 0;

Int(Int(y/(1+x^2), x = A .. ``), y = `` .. ``) = 0

En effet, A est symétrique par rapport à (Oy) et

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

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

question c)

> Dom:={abs(x)<=1,abs(y)<=1};

Dom := {abs(x) <= 1, abs(y) <= 1}

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

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

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

[Plot]

> Doubleint('f'(x,y),x,y,A)= 2*Doubleint('f'(x,y),x,y,Ap);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 2*Int(Int(f(x, y), x = Ap .. ``), y = `` .. ``)

En effet, A est symétrique par rapport à la première bissectriece: y=x et

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

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

Plus particulièrement en notant Ap et Am les parties de A au dessu et en dessous de la première bissectrice

> Doubleint('f'(x,y),x,y,A)=Doubleint('f'(x,y),x,y,Ap)+Doubleint('f'(x,y),x,y,Am);

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = Int(Int(f(x, y), x = Ap .. ``), y = `` .. ``)+Int(Int(f(x, y), x = Am .. ``), y = `` .. ``)

> Or par changement de variable u=y,v=x;

u = y, v = x

> Doubleint('f'(u,v),u,v,Am)=Doubleint('f'(x,-y),x,y,Ap),"=",Doubleint('f'(x,y),x,y,Ap);

Int(Int(f(u, v), u = Am .. ``), v = `` .. ``) = Int(Int(f(x, -y), x = Ap .. ``), y = `` .. ``),

Voici le domaine d'intégration Ap

> inequal({y>=x,x>=-1,x<=1,y<=1},x=-1..1,y=-1..1,optionsexcluded=(color=white));

[Plot]

> Doubleint('f'(x,y),x,y,Ap)=Doubleint(f(x,y),x=-1..y,y=-1..1) ,"=",value(Doubleint(f(x,y),x=-1..y,y=-1..1));

Int(Int(f(x, y), x = Ap .. ``), y = `` .. ``) = Int(Int(abs(x-y), x = -1 .. y), y = -1 .. 1),

> Doubleint('f'(x,y),x,y,A)=8/3;

Int(Int(f(x, y), x = A .. ``), y = `` .. ``) = 8/3

>

question d)

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

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

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

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

> implicitplot(Dom,x=-1..1,y=-1..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

> inequal({r>=0,r<=1,theta<=Pi/4,theta>=0},r=-1..1,theta=-1..1,optionsexcluded=(color=white));

[Plot]

> Doubleint(f(x,y),x,y,A)=  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..1,theta=0..Pi/4) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*abs(r),r=0..1,theta=0..Pi/4) );

Int(Int(x*y*(x^2+y^2)^(1/2), x = A .. ``), y = `` .. ``) = Int(Int(r^4*cos(theta)*sin(theta)*csgn(r), r = 0 .. 1), theta = 0 .. 1/4*Pi),

>

question e)

> Dom:={x>=0,y>=0,1<=x^2+y^2, x^2+y^2<=4};

Dom := {0 <= y, 1 <= x^2+y^2, 0 <= x, x^2+y^2 <= 4}

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

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

> implicitplot(Dom,x=-3..3,y=-3..3,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

> inequal({r>=1,r<=2,theta<=Pi/2,theta>=0},r=-1..3,theta=-1..2,optionsexcluded=(color=white));

[Plot]

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

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

>

question f)

> Dom:={ x^2/a^2+y^2/b^2<=1};

Dom := {x^2/a^2+y^2/b^2 <= 1}

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

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

> a:=1:b:=2:implicitplot(Dom,x=-3..3,y=-3..3,thickness=3);a:='a':b:='b':

[Plot]

Changement de coordonnées affine

> u=a*x,v=b*y;

u = a*x, v = b*y

> Doubleint('f'(u,v),u,v,A)= Doubleint('f'(a*x,b*y)*abs(a*b),x,y,C);

Int(Int(f(u, v), u = A .. ``), v = `` .. ``) = Int(Int(f(a*x, b*y)*abs(a*b), x = C .. ``), y = `` .. ``)

Où le domaine d'intégration est le cercle trigo C

> a:=1:b:=1:implicitplot(Dom,x=-3..3,y=-3..3,thickness=3);a:='a':b:='b':

[Plot]

Changement de coordonnées polaires

> Doubleint('f'(a*x,b*y),x,y,C)= Doubleint('f'(a*r*cos(theta),b*r*sin(theta))*abs(r),r,theta,Cpol);

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

domaine d'intégration polaire

> inequal({r>=0,r<=1,theta<=2*Pi,theta>=0},r=-1..2,theta=-1..7,optionsexcluded=(color=white));

[Plot]

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

Int(Int(x^2+y^2, x = A .. ``), y = `` .. ``) = Int(Int(-(-a^2*cos(theta)^2-b^2+b^2*cos(theta)^2)*r^3, r = 0 .. 1), theta = 0 .. 2*Pi),

> Doubleint('f'(u,v),u,v,A)=a*b/4*Pi(a^2+b^2);

Int(Int(f(u, v), u = A .. ``), v = `` .. ``) = 1/4*a*b*Pi(b^2+a^2)

>