exo25.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 25

question a)

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

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

> f:=(x,y)-> (x^3+y^3);

f := proc (x, y) options operator, arrow; x^3+y^3 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 quart de cercle trigo C

> a:=1:b:=1:implicitplot(Dom,x=-2..2,y=-2..2,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 = C .. ``), y = `` .. ``) = Int(Int(f(a*r*cos(theta), b*r*sin(theta))*abs(r), r = Cpol .. ``), theta = `` .. ``)

domaine d'intégration polaire

> inequal({r>=0,r<=1,theta<=Pi/2,theta>=0},r=-1..2,theta=-1..2,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..Pi/2) ,"=",value(  Doubleint(simplify(f(a*r*cos(theta),b*r*sin(theta))*r),r=0..1,theta=0..Pi/2));

Int(Int(x^3+y^3, x = A .. ``), y = `` .. ``) = Int(Int(-r^4*(-b^3*sin(theta)+b^3*sin(theta)*cos(theta)^2-a^3*cos(theta)^3), r = 0 .. 1), theta = 0 .. 1/2*Pi),

> Doubleint('f'(u,v),u,v,A)=2*a*b/15*(a^3+b^3);

Int(Int(f(u, v), u = A .. ``), v = `` .. ``) = 2/15*a*b*(a^3+b^3)

question b)

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

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

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

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

> implicitplot(Dom,x=0..2,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 Apol

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

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

> implicitplot(Dpol,r=0..2,theta=0..1.55,thickness=3);

[Plot]

> Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol)=  Doubleint(f(r*cos(theta),r*sin(theta))*abs(r),r=0..2*cos(theta),theta=0..Pi/2) ,"=",value(   Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r=0..2*cos(theta),theta=0..Pi/2) );

Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``) = Int(Int(r^3*cos(theta)^2*sin(theta)*abs(r), r = 0 .. 2*cos(theta)), theta = 0 .. 1/2*Pi),

question c)

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

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

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

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

> implicitplot(Dom,x=0..2,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 Apol

> Dpol:={-Pi/4<=theta,theta<=Pi/4,1/cos(theta)<=r,r<=2*cos(theta)};

Dpol := {theta <= 1/4*Pi, 1/cos(theta) <= r, -1/4*Pi <= theta, r <= 2*cos(theta)}

> implicitplot(Dpol,r=0..2,theta=-0.77..0.77,thickness=3);

[Plot]

> Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol)=  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=1/cos(theta)..2*cos(theta),theta=-Pi/4..Pi/4) ,"=",value(  Doubleint(f(r*cos(theta),r*sin(theta))*r,r=1/cos(theta)..2*cos(theta),theta=-Pi/4..Pi/4) );

Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``) = Int(Int(1/r^3, r = 1/cos(theta) .. 2*cos(theta)), theta = -1/4*Pi .. 1/4*Pi),

question d)

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

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

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

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

> Dpol:={0<=theta,theta<=Pi/6,2*sin(theta)<=r,r<=1};

Dpol := {2*sin(theta) <= r, 0 <= theta, theta <= 1/6*Pi, r <= 1}

> implicitplot(Dpol,r=0..1,theta=0..0.52,thickness=3);

[Plot]

> Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol)=  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=2*sin(theta)..1,theta=0..Pi/6) ,"=",value(  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=2*sin(theta)..1,theta=0..Pi/6) );

Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``) = Int(Int(csgn(r)*r^2, r = 2*sin(theta) .. 1), theta = 0 .. 1/6*Pi),

>

>

> question e)

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

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

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

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

> implicitplot(Dom,x=-1..2,y=-2..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 = `` .. ``)

domaine d'intégration polaire Apol

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

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

> implicitplot(Dpol,r=0..2,theta=-1.52..1.52,thickness=3);

[Plot]

> Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol)=  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..2*cos(theta),theta=-Pi/2..Pi/2) ,"=",value(  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..2*cos(theta),theta=-Pi/2..Pi/2));

Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``) = Int(Int((2*cos(theta)^2-1)*r^3, r = 0 .. 2*cos(theta)), theta = -1/2*Pi .. 1/2*Pi),

question f)

> Dom:={ x^2+y^2<=R*x};

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

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

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

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

[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 Apol

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

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

> R:=2:implicitplot(Dpol,r=0..2,theta=-1.52..1.52,thickness=3);R:='R':assume(R>0):

[Plot]

> Doubleint('f'(r*cos(theta),r*sin(theta))*abs(r),r,theta,Apol)=  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..R*cos(theta),theta=-Pi/2..Pi/2) ,"=",value(  Doubleint(simplify(f(r*cos(theta),r*sin(theta))*r),r=0..R*cos(theta),theta=-Pi/2..Pi/2));

Int(Int(f(r*cos(theta), r*sin(theta))*abs(r), r = Apol .. ``), theta = `` .. ``) = Int(Int((R^2-r^2)^(1/2)*r, r = 0 .. R*cos(theta)), theta = -1/2*Pi .. 1/2*Pi),

>