4. Xosmas integralnin geometrik masalalarga tadbiqi

Yechish. Yuz elimenti: .
Izlanayotgan yuza qiymati integrallash chegaralari cheksiz bo`lgan xosmas integralga teng:

> restart;
> with(plots): f:=x->8/(x^2+4):
> plot({f(x)}, x=-6..6, y=0..2,color=red, style=line, thickness=2, title=`YUZA`);

> XI1:=int( a^3/(x^2+a^2), x=-infinity..infinity );

> a:=2:XI1;
7-misol. strofoida va uning asimptotasi bilan chegaralangan yuzani hisoblang.
Yechish. Yuz elimenti: .
Izlanayotgan yuza qiymati uzlykli funktsiyadan olingan xosmas integralga teng:

Integralostidagi funktsiya x=2a nuqtada uzilishga ega. Bu integralda
x=2asin²t , dx=4a sint cost, a≤x≤2a dan π/4≤t≤ π/2
ga o`tib quyidagi yechimni topamiz:

Strofoida grafigini uning parametrik tenglamasi x=1+sinφ, y=(1+sinφ) sinφ/cosφ asosida quramiz:
> with(plots):
> plot([1*(1+sin(t)), 1*(1+sin(t))*sin(t)/cos(t), t=0..2*Pi], 0..4, -4..4, color=blue,thickness=2,title=`Strofoida`);



> XI3:=2*int((x-a)*sqrt(x/(2*a-x)),x=a..2*a);

∫



> value(%);

∫



> a:=1:XI3;
8-misol. (x>1) egri chizuq cheksiz tarmog`ining Ox o`qi atrofida aylanishdan hosil bolgan jisim xajmini hisoblang.
Yechish. Aylanish xajmi elimenti: .
Izlanayotgan jism qiymatini chegarasi bo`lgan quyidagi integralga teng:

1)grafigini quyidagich quramiz:
> restart;
> with(plots):
Warning, the name changecoords has been redefined
> implicitplot(y=2*(1/x-1/(x^2)), x=0..6, y=-1..1,color= blue, thickness=2);

2)jisim xajmini 2 xil usulda hisoblaymiz.