A niq integralning geometriyaga tatbiqlari Reja

> with(plots):
> implicitplot(r=abs(cos(phi)), r=-1..1, phi =0..2*Pi, coords=polar,thickness=2,title=`Bernuli lyumniskatasi`);



12.8 –rasm.
> restart;
> with(IntegrationTools):
S := Int((cos(phi)^2)/2, phi=0..2*Pi);

> value(%);
> evalf(S,5); 1.5708
15-misol. Qutb koordinatalari sistemasida =a aylana va =2acos3 uch yaproq chiziqlari bilan chegaralangan aylana tashqarisida hosi bo’lgan yuzalarni hisoblang.
Yechish. =2acos3 chiziq T=2π/3 davr bilan [-π, π] da chiziq quriladi. Unda
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: