Aniq integralni geometriya va mexanikaga tadbiqlari

cos3≥0 ekanidan birinch bo’lagi [-π/6, π/6] oraloqda joylashgan. Bu bo’lakning aylana bilan kesishish niqtalari 2acos3 =a tenglamadan ₁₌-π/9, ₂₌π/9 bo’lasi.



> restart;
> with(plots):
polarplot([1,2*cos(3*t)],t=0..2*Pi,color= [blue,red], thickness=2);

> p1:=phi->a; p2:=phi->2*a*cos(3*phi);

> S3:=(1/2)*int(p2(phi)^2-p1(phi)^2, phi=-Pi/.Pi/9);

> a:=1:S3:value(%);
> evalf(%); 0.6377409851
Tekisligida parametrik tenglamalar bilan berilgan chiziq bilan chegaralgan tekis shakl yuzini hisoblash. Aytaylik, Dekart koordinatalar sistemasida t parametrning [α, β] kesmada monoton o‘zgarishga mos x=x(t), y=y(t) parametrik tenglamalar bilan berilgan chiziq bilan chegaralgan tekis shakl yuzini quyidagi formulalar bilan hisoblaymiz:

16-misol. x²+y²=9 aylana va 4x²+9y²=36 ellips bilan chegaralangan birjinsli shaklning 1-chorakdagi bo’lagining yuzasini hisoblang (26- rasmga qarang).
Yechish.
1) x²+y²=9 dan:
4x²+9y²=36 dan:

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

> s:=int(f2(x)-f1(x),x=0..3);
2)berilgan ellipsning parametrik tenglamasiga o’tib yuzani hisoblaymiz:
4x²+9y²=36 dan:

3)berilgan aylananing parametrik tenglamasiga o’tib yuzani hisoblaymiz:
x²+y²=9 dan:

4)Izlanayotgan yuza:
> restart;with(plots):
> x2:=t->3*cos(t);y2:=t->3*sin(t);

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

> s1:=plot([x1(t),y1(t), t=0..Pi/2],thickness=2);

> s2:=plot([ x2(t),y2(t), t=0..Pi/2],thickness=2);

> display({s2,s1}, axes=boxed, title=`YUZA`);

> 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;
17-misol. astroida-chizig’i bilan chegaralangan egri chiziqli yassi yuzani hisoblash.
1) astroida grafigini quramiz:
> with(plots):
> plot([1*cos(t)^3,1*sin(t)^3, t=0..2*Pi],thickness=2);

2)berilgan astroidaning parametrik tenglamasi x=acos³(t), y=asin³(t) ga asosan uning yuzani hisoblash:

> restart;
> x:=t->a*cos(t)^3;y:=t->a*sin(t)^3;

> s:=4*(1/2)*int(x(t)*diff(y(t),t)-y(t)*diff(x(t),t), t=0..Pi/2);

Parallel kesimlarining yuzi bo‘yicha jismning hajmini hisoblash

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