Xosmas integral Reja

> with(IntegrationTools):

XI2 := Int(1/sqrt(abs(x*(x^2-1))), x=-infinity..infinity);



> Split(XI2, [-1, 0, 1]);



> XI2:=value(%);



> evalf(XI2,5);



5-misol. xosmas integralning yaqinlashuvchiligi tekshirilsin (R).

Yechish. Bu yerda (-;1), =1 va (1;) bo`lgan uch holni ajratamiz.

1. 
   1 bo`lsin, u holda
   

oxirgi limit  <1 bo`lganda mabjuddir va uning qiymati ga tengdir, ya`ni

- xosmas integral yaqinlasuvchi.

Agar >1 bo`lsa,

,

ya`ni, - xosmas integral uzoqlashuvchi.

2) , ya`ni xosmas integral uzoqlashuvchi ekan.

Demak, -xosmas integral <1 bo`lganda yaqinlashuvchi, 1 bo`lganda esa uzoqlashuvchidir.

Xosmas integralnin geometrik masalalarga tadbiqi
6-misol. Anyezi chziq zulfi va abtsissalar o`qi orasida joylashgan yuzani hisoblang.

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.

a) formula bo`yicha:

> XI4:=4*Pi*Int((1/x-1/(x^2))^2,x=1..infinity);



> XI4:=4*Pi*int((1/x-1/(x^2))^2,x=1..infinity);

b) VolumeOfRevolution buyrug`i bo`yicha jisim xajmini [1,6] dagi qismi:

> restart; with(plots): with(Student[Calculus1]):

> VolumeOfRevolution((x-1)/x^2,x=1..6,output=plot);



> VolumeOfRevolution(2*(x-1)/x^2,x=1..6, output=integral);



> value(%);