Mavzu : Aniq integralning geometriya va fizika masalalariga tadbiqlari Reja

11-misol. y=3x-x² va y=-x chiziqlar bilan chegaralangah yuza hisoblansin.
Yechish. Ciziqlarning kesishish nuqtalar kordinatalarini topish uchun tenglamalarini sistema qilib yechmiz va : x=0, y=0 ; x=4, y=-4. Bu holda yuza:

> restart;with(plots):
> f2:=x->3*x-x^2: f1:=x->-x:
> plot({f2(x),f1(x)}, x=-2..5, y=-5..3,color=[red,blue], style=line, thickness=2, title=`YUZA`);



> Int(f2(x)-f1(x), x=0..4)=int(f2(x)-f1(x), x=0..4);

12-misol. y=2-x^2 va y³=x² chiziqlar bilan chegaralangah yuza hisoblansin.
Yechish. Ciziqlarning kesishish nuqtalar kordinatalarini topish uchun tenglamalarini sistema

> restart; with(plots):with(Student[Calculus1]):
> f2:=x->2-x^2: f1:=x->(x^2)^(1/3):
> plot({f2(x),f1(x)}, x=-2..2, y=0..2,color=[red,blue], style=line, thickness=2, title=`YUZA`);
Int(f2(x)-f1(x), x=-1..1)=int(f2(x)-f1(x), x=-1..1);



13-misol. chiziqlar bilan chegaralangah yuza hisoblansin.
Yechish. Ciziqlarning kesishish nuqtalar kordinatalarini topish uchun tenglamalarini sistema qilib yechmiz va :
dan x=2, y=4 ; dan x=4, y=8
Bundan hosil bo’lgan yuzaning Ox o’qdagi proyektsiyasi [0,4] da bo’adi. Yuzani hisoblashda uni dan ga o’yish nuqtasining abtsissasi x=2 to’g’ri chiziq bilan ikkiga bolamiz

> restart; with(plots):with(Student[Calculus1]):
> f1:=x->x^2: f2:=x->(x^2)/2: f3:=x->2*x:
> plot({f1(x),f2(x),f3(x)}, x=-0..5, y=0..10, color=[red,blue,green],style=line, thickness=2, title=`YUZA`);


> S1:=Int(f1(x)-f2(x), x=0..2)=int(f1(x)-f2(x), x=0..2);

> S2:=Int(f3(x)-f2(x), x=2..4)=int(f3(x)-f2(x), x=2..4);