Maple dasturi yordamida aniqmas va aniq integrallarni hisoblash

>Doubleint(x^2-y^2+2,y=-sqrt(4-x^2)..sqrt(4-x^2),x=-2..2);

2

 
_ _
 


x²  y²  2 dy dx

>value(%);

-2 _

8 

>Doubleint(abs(x)+abs(y),x=y..y^2,y=1..2);

2 y²
_ _{ x }


y dx dy

>value(%);

₁ _y


67


20

>f:=piecewise(x+y<1,x^3+y^3,x+y>=1,x-y);

f := {
x³  y³ x  y
x  y  1 1  x  y

>Doubleint(f,x=-3..5,y=-6..9);

9 5
_ _
x³  y³
x  y  1

 
  ^{
 
x  y
dx dy
1  x  y

>value(%);

-6 -3
-11996

5

2. 
   Takroriy integrallarni hisoblash uchun int buyrug’idan ichma – ich foydalanish yetarli. Masalan,
   

• Int(Int(x+2*y,x=1..2),y=0..3)=int(int(x+2*y,x=1..2),y=0..3);

3 2 ₂₇
^_ ^_ x  2 y dx dy  ₂
0 1

• Int(Int(Int(x+y+z,x=-1..1),y=0..1),z=-1..1)=

int(int(int(x+y+z,x=-1..1),y=0..1),z=-1..1);
1 1 1

  
^{  } x  y  z dx dy dz  2
  
-1 0 -1
Umuman olganda, Maple dasturi ikki karrali integrallarni takroriy integralga keltirilgan holdagisini hisoblay oladi. Shuning uchun ikki karrali integrallarni hisoblashda yo I punktdagi buyruqdan yoki II punktdagi qoidadan foydalanish mumkin. Har ikkala holda ham natija bir xil bo’ladi.

>restart;

>with(student):

>Doubleint(x+y*exp(x),y=-x..x^3,x=-1..3);

3

 
 
_ _
x3
x  y e^x dy dx

>value(%);

-1 x

⁸⁷²  128 e³  976 e(-1 )
15

• Int(Int(x+y*exp(x),y=-x..x^3),x=-1..3)=int(int(x+y*exp(x),y=- x..x^3),x=-1..3);

3

 
 
_ _
x3
x  y e^x dy dx 
872

15
 128e³  976e^{( -1 )}

-1 x
Ikki karrali integralda ham o’zgaruvchilarni almashtirish imkoniyati Maple dasturida mavjud bo’lib, bu amal changevar funksiyasi orqali amalga oshiriladi. Bunga doir misollar keltiramiz:

>restart;

>with(student):

>Doubleint(f(x^2+y^2),x,y);

f( x²  y² ) dx dy

>changevar({x=r*cos(t),y=r*sin(t)},%,[r,t]);

f( r² )
r dr dt

>Doubleint(f(x,y),y=alpha*x..beta*x,x=a..b);

b
_ _
 x
f( x, y ) dy dx

^a ^ x

>changevar({u=x,v=y/x},%,[u,v]);

_^f( u, v u ) u du dv


>Doubleint(f(sqrt(1-x^2/a^2-y^2/b^2)),x,y);

__{ }
x² y² _

^f
__ ^
1  _a2  _{b2 } dx dy

>changevar({x=a*r*cos(phi),y=b*r*sin(phi)},%,[r,phi]);


^_^_f( ) a

r b dr d

Keltirilgan misollarning ikkinchisidan xulosa qilish mumkinki, Maple dasturi ikki karrali aniq integrallarda o’zgaruvchini almashtirganda integrallash oralig’ini aniqlamasdan natijani ekranga chiqaradi.

3. 
   Uch karrali integrallarni hisoblash uchun
   

>Tripleint(f(x,y,z),x,y,z);
>Tripleint(f(x,y,z),x=a..b,z=e..f,y=c..d);

buyruqlaridan foydalaniladi. Bunda ravshanki, yangi z parametr – uchinchi o’zgaruvchi paydo bo’ladi va qolgan parametr va buyruqlarning vazifasi yuqoridagi kabi. Misollar bilan tanishamiz:

>restart;

>with(student):

>Tripleint(f(x,y,z),x,y,z);

^_^_f( x, y, z ) dx dy dz

>Tripleint(f(x,y,z),x=a..b,z=e..f,y=c..d);

d f b
^_ ^_ f( x, y, z ) dx dz dy
_c _e _a

>Tripleint(x^2+y^2+z^2,x,y,z);


_^_^x²  y²  z² dx dy dz


>value(%);

¹ x³ y z  ¹ y³ x z  ¹ z³ x y

3 3 3

>Tripleint(x-y+z,x=-1..1,y=-2..1,z=-3..3);

3 1 1
^_ ^_ x  y  z dx dy dz

>value(%);

^-3^-2^-1


18

>Tripleint(x*y*z,z=x*y..y+x,y=x..x+1,x=0..1);

1 x  1
_ _
y  x
x y z dz dy dx

>value(%);

₀ _x
^x y

9199

10080

>Tripleint(f(x^2+y^2-z^2),x,y,z);


_^_^f( x²  y²  z² ) dx dy dz


>changevar({x=r*cos(xi),y=r*sin(xi),z=h},%,[xi,r,h]);


^_^_^_f( h²  r² )

r d dr dh

>Tripleint(f(sqrt(x^2+y^2+z^2)),x,y,z);


^_^_^_f( ) dx dy dz


>changevar({x=w*cos(u)*cos(v),y=w*sin(u)*cos(v),z=w*sin(v)},%,[u,v, w]);


^_^_^_f( w ) w ²

cos( v )
du dv dw

Ikki karrali integrallarda bo’lgani kabi uch karrali integrallarda ham Maple dasturi faqat takroriy integralga keltirilgan uch karrali integrallarni hisoblash imkoniyatiga ega. Shunday bo’lgani uchun takroriy integralga keltirilgan uch karrali integrallarni hisoblash uchun int funksiyasidan ichma – ich foydalanish yetarli.

• Int(Int(Int(x+y+z,x=-1..1),y=0..1),z=-1..1)=

int(int(int(x+y+z,x=-1..1),y=0..1),z=-1..1);
1 1 1

  
^{  } x  y  z dx dy dz  2
  
-1 0 -1

• Int(Int(Int(x+y+z,x=-1..1),y=0..2),z=-3..2);

2 2 1

  
^{  } x  y  z dx dy dz
  
-3 0 -1

• int(int(int(x+y+z,x=-1..1),y=0..2),z=-3..2);

10

• Int(Int(Int(sin(x)*sin(y)*sin(z),x=0..Pi),y=Pi..2*Pi),z=0..Pi);

 2  

 


^{  } sin( x ) sin( y ) sin( z ) dx dy dz
  

• value(%);

0  0
-8

• Int(Int(Int(x-y*z,x),y),z)=int(int(int(x-y*z,x),y),z);

_^_^x  y z dx dy dz  ¹ x² y z  ¹ y² z² x
 ^{2 4}

• with(student):Tripleint(x-y*z,x,y,z);


^x  y z dx dy dz


• value(%);

¹ x² y z  ¹ y² z² x

2 4

Takroriy integralga keltrilgan ikki va uch karrali xosmas integrallarni hisoblashda ham Doubleint va Tripleint funksiyalaridan foydalaniladi. Bunga doir quyida misollar keltirib o’tilgan va ularni mustaqil tahlil eting:

>restart;

>with(student):

>Doubleint((x^2-y^2)/(x^2+y^2)^2,x=1..infinity,y=1..infinity);

 
_ _
 
 


x²  y²
( x²  y² )


₂ dx dy

>value(%);

₁ ₁


1 _
4

>Doubleint((x^2-y^2)/(x^2+y^2)^2,y=1..infinity,x=1..infinity);

 
_ _
 
 


x²  y²
( x²  y² )


₂ dy dx

>value(%);

₁ ₁
1
 ₄ 

>Doubleint(1/(x+y)^4,y=1-x..infinity,x=0..1);

1 
_ _

 
 
 
¹ dy dx
( x  y )⁴

>value(%);

0 1  x

1

3

>Doubleint(1/sqrt(1-x^2-y^2),y=-sqrt(-x^2+1)..sqrt(1-x^2),x=-1..1);

1

_ _

 
 
 
¹ dy dx

>value(%);

-1 _

2 

>Tripleint(1/sqrt(x*y*z),x=0..1,y=0..1,z=0..1);

1 1 1

  

  
^{   1} dx dy dz
  

>value(%);

0 0 0
8

>limit(Doubleint(sin(x^2+y^2),x=-n..n,y=-n..n),n=infinity);

lim
n n






^{ } sin( x²  y² ) dx dy

>value(%);

n  
^n^n



>Doubleint(x*y*exp(-(x^2+x*y+y^2)),x=-infinity..infinity,y=- infinity..infinity);

 
_ _
 
_{x y e}( x²  x y  y² ) _{dx dy}