exo21.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 21

question b)

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

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

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

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

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

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=0..1-x,x=0..1) ,"=",value(Doubleint(f(x,y),y=0..1-x,x=0..1));

Int(Int(x*y*(x+y), x = A .. ``), y = `` .. ``) = Int(Int(x*y*(x+y), y = 0 .. 1-x), x = 0 .. 1),

question c)

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

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

> f:=(x,y)->ln(1+x+y);

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

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

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=0..1-x,x=0..1) ,"=",value(Doubleint(f(x,y),y=0..1-x,x=0..1));

Int(Int(ln(1+x+y), x = A .. ``), y = `` .. ``) = Int(Int(ln(1+x+y), y = 0 .. 1-x), x = 0 .. 1),

question d)

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

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

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

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

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

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=0..(1-x)/(1+x),x=0..1) ,"=",value(Doubleint(f(x,y),y=0..(1-x)/(1+x),x=0..1));

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

>

question e)

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

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

> f:=(x,y)->1/(x+y)^4;

f := proc (x, y) options operator, arrow; 1/(x+y)^4 end proc

> inequal(Dom,x=-1..5,y=-1..5,optionsexcluded=(color=white));

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=1..4-x,x=1..3) ,"=",value(Doubleint(f(x,y),y=1..4-x,x=1..3));

Int(Int(1/(x+y)^4, x = A .. ``), y = `` .. ``) = Int(Int(1/(x+y)^4, y = 1 .. 4-x), x = 1 .. 3),

question f)

> Dom:={x>=y-1,y>=0,x<=4-2*y};

Dom := {0 <= y, x <= 4-2*y, y-1 <= x}

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

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

> inequal(Dom,x=-1..5,y=-1..3,optionsexcluded=(color=white));

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=0..x+1,x=-1..2/3) + Doubleint(f(x,y),y=0..2-x/2,x=2/3..4) ,"=",value(Doubleint(f(x,y),y=0..x+1,x=-1..2/3) + Doubleint(f(x,y),y=0..2-x/2,x=2/3..4));

Int(Int(x, x = A .. ``), y = `` .. ``) = Int(Int(x, y = 0 .. 1+x), x = -1 .. 2/3)+Int(Int(x, y = 0 .. 2-1/2*x), x = 2/3 .. 4),

question g)

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

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

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

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

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

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=0..1-x^2,x=0..1) ,"=",value(Doubleint(f(x,y),y=0..1-x^2,x=0..1));

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

>

question h)

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

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

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

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

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

[Plot]

> Doubleint(f(x,y),x,y,A)= Doubleint(f(x,y),y=x^2..sqrt(x),x=0..1) ,"=",value(Doubleint(f(x,y),y=x^2..sqrt(x),x=0..1));

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

>