Reja: Statik moment. Og`irlik markazi

> restart; with(plots):
> f:=x->2*x-x^2:
> plot(f(x),x=0..2,color=red,style=line, thickness=2, title=`Yuza`);

> restart;
> Int(2*x-x^2,x=0..2)=int(2*x-x^2,x=0..2);

> s:=int(2*x-x^2,x=0..2);
> Int((2*x-x^2)^2/2,x=0..2)=int((2*x-x^2)^2/2,x=0..2);

> Mx:=int((2*x-x^2)^2/2,x=0..2);
> My:=Int(x*(2*x-x^2),x=0..2)=int(x*(2*x-x^2),x=0..2);

> My:=int(x*(2*x-x^2),x=0..2);
> My/s; Mx/s; 1
41-misol. x²+y²=9 aylana va 4x²+9y²=36 ellips bilan chegaralangan birjinsli shaklning 1-chorakdagi bo`lagining og`irlik markazi topilsin (26- rasmga qarang).




Masalada ko`rsatilgan shakilning og`irlimk markazi nuqtada. Bu misoldan ko`rinadiki, og`irlik markazi bu shaklning simmetriya o`qidadir.
> restart;with(plots):
> implicitplot([x^2+y^2=9,4*x^2+9*y^2=36], x=0..3, y=0..3, color= [blue,red], thickness=2);

26- rasm


> f1:=x->2*sqrt(9-x^2)/3;f2:=x->sqrt(9-x^2);

> My:=int(x*(f2(x)-f1(x)),x=0..3);
> Mx:= int((f2(x)^2-f1(x)^2)/2,x=0..3);
> s:=int(f2(x)-f1(x),x=0..3);
> Xc:=My/s; Yc:=Mx/s;
Egri chiziq tenglamalari parametrik ko`rinishda bo`lganda yuqoridagi masalani yechamiz.
> x2:=t->3*cos(t);y2:=t->3*sin(t);

> x1:=t->3*cos(t);y1:=t->2*sin(t);

> s1:=int(x1(t)*diff(y1(t),t),t=0..Pi/2);
> s2:=int(x2(t)*diff(y2(t),t),t=0..Pi/2);
> s:=s2-s1;
> My:=int((x1(t)*y1(t)*diff(x1(t),t)-x2(t)*y2(t)*diff(x2(t),t)), t=0..Pi/2);