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International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645

The Method of Repeated Application of a Quadrature Formula of Trapezoids and Rectangles to Determine the Values of Multiple Integrals Nazirov Sh. A. Tashkent University of Information Technologies, Tashkent, Uzbekistan

Abstract: The work deals with the construction of multi-dimensional quadrature formulas based on the method of repeated application of the quadrature formulas of rectangles and trapezoids, to calculate the multiple integrals’ values. We’ve proved the correctness of the quadrature formulas.

Keywords: multi-dimensional integrals, cubature formulas, re-use method. I. INTRODUCNION The paper is devoted to developing the multiple cubature formulas for the values of n-fold integrals calculation by means of methods of repeated application of, quadrature formulas of trapezoid and rectangles. According to mathematical analysis multiple integrals can be calculated by recalculating single integrals. The essence of this approach is the following. Let the domain of integration is limited to single-valued continuous curves [1]

y   ( x), y   ( x) ( ( x)   ( x)) and two vertical lines x  a, x  b (Fig.1).

Placing the known rules in the double integral

I

 f ( x, y)dxdy

(1)

( D)

limits of integration, we obtain



f ( x, y )dxdy 

( D)

b

 ( x)

a

 ( x)

 dx

f ( x, y )dy.

Let  ( x)

F ( x) 

 

(2)

f ( x, y )dy.

( x)

Then b

 f ( x, y)dxdy   F ( x)dx. ( D)

(3)

a

y

y   (x)

y   (x) 0

а

b

x

Fig.1 Applying to a single integral in the right-hand side of (3), one of the quadrature formulas, we have



( )

where -

n

f ( x, y )dxdy   Ai F ( xi ),

(4)

i 1

xi  [a, b] (i  1,2,..., n) and Ai are some constant coefficients. In turn, the value

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F ( xi ) 

International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645

 ( xi )

f ( xi , y )dy

 ( xi )

can also be found on some quadrature formulas mi

F ( xi )   Bij f ( xi , y j ), j 1

Bij iy - appropriate constants.

where

From (4) we obtain n

mi

 f ( x, y)dxdy   A B i

i 1 j 1

( D)

ij

f ( xi , y j ),

(5)

Ai and Bij iy - are known constants.

where

Geometrically, this method is equivalent to the calculation of the volume I, given by the integral (2) by means of cross-sections. For cubature formulas of the type (3), the general comments, relating to the computation of simple integrals, are valid with appropriate modifications. In three dimensions (3) has the form



mi

n

ti

f ( x, y, z )dxdydz   Ai Bij Cij ,k f ( xi , y j , z k ) . i 1 j 1 k 1

( D)

Now these results are generalized to compute the values of multiple integrals

 ...  f ( x , x 1

( D) n1

2

, x k )dx1 dx 2 ...dx k 

nk

n2

  ... Ai(11) Ai(1i22)i3 Ai(1i32)i3 ... Ai(1ik2)ik f ( x1i , x 2i ,..., x ki ), i1 1 i2 1

Where

in 1

(1) i1

A , Ai(1i22) ,..., Ai(1ik2)...ik - are known constants.

This approach can be applied to calculate the approximate value of multiple integrals using appropriate quadrature formulas. In this case, we apply the quadrature formula of trapezoid.

II. ALGORITHM DISCRIPTION Theorem 1. Let the function

y  f ( x1 , x2 ,..., xn ) Is continuous in a bounded domain D. Then the formula for the approximate values of the n-fold integral calculating

J n   ...  f ( x1 ,..., xn )dx1dx2 ,..., dxn ,

(6) based on the trapezoidal

( D)

rule has the form n1 1 n2  n3 1

nn 1 1

p1 0 p2 0 p3 0

pn 0i1 0 i2 0

1

1

J n   hi    ...  ... f p1 i1 , p2 i2 ,...,pn in i 1

in 0

Here D is n-dimensional domain of integration of the form

ai  xi  Ai , hi 

i  1, n ;

(7)

Ai  ai 2

and it is believed that each interval (5), is respectively, divided into

n1 , n2 ,..., nn parts.

Proof. Approximate formula for calculating the values of the one-dimensional integral in this case is

J1 

x1

f ( x)dx 

x0

where,

hx 

hx  f ( x0 )  f ( x1 )  hx 2 2

1

f i 0

i

,

(8)

ba , n - the number of the interval divisions into n parts. In (4) n = 1 . n

Now we define the approximate calculation formula for double integrals

J2 

 f ( x, y)dxdy

(9)

( D)

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International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 where D is the two-dimensional area of integration. It is generally believed that the region of integration in this case is within the rectangle, that is, a  x  b, c  y  d . Then we rewrite the integral (7) in the form b

d

J 2   dx  f ( x, y )dy . a

(10)

c

Using the trapezoidal rule to calculate the inner integral, we obtain

J2 

b hy  b  f ( x , y ) dx  f ( x, y )dx  .   2 a a 

(11)

Now to each integral we use the trapezoid formulas and the result is

h y  hx hx    f ( x0 , y 0 )  f ( x1 , y 0 )   f ( x0 , y1 )  f ( x1 , y1 )  2 2 2  (12) hx h y hx h y 1 1  f ( x0 , y0 )  f ( x1 , y0 )  f ( x0 , y1 )  f ( x1 , y1 )   f ( xi , y j ), 4 4 i 0 j 0 d c where h y  . m J2 

Similarly, for the approximate calculation of the values of the triple integral we obtain the following formula: ABC

J 3     f ( x, y, z )dxdydz 

hx h y hz

a b c

8

 f ( x0 , y0 , z 0 )  f ( x0 , y0 , z1 )  f ( x0 , y1 , z 0 ) 

 f ( x0 , y1 , z1 )  f ( x1 , y 0 , z 0 )  f ( x1 , y 0 , z1 )  f ( x1 , y1 z 0 )  f ( x1 , y1 z 0 )  f ( x1 , y1 , z1 )  (13) hx h y hz

1

8 where

1

1

 f ( x , y i 0 j 0 k 0

hz 

i

j

, z k ),

C c . p

The four-dimensional integral is defined with 16 members: ABCD hx h y hz hu  f ( x0 , y0 , z 0 , u 0 )  f ( x0 , y0 , z 0 , u1 )  f ( x0 , y0 , z1 , u 0 )  J 4      f ( x, y, z, u )dxdydzdu  16 a b c d

 f ( x0 , y 0 , z1 , u1 )  f ( x0 , y1 , z 0 , u 0 )  f ( x0 , y1 , z 0 , u1 )  f ( x0 , y1 , z1 , u 0 )  f ( x0 , y1 , z1 , u1 ) 

(14)

 f ( x1 , y 0 , z 0 , u 0 )  f ( x1 , y 0 , z 0 , u1 )  f ( x1 , y 0 , z1 , u 0 )  f ( x1 , y 0 , z1 , u1 )  f ( x1 , y1 , z 0 , u 0 )   f ( x1 , y1 , z 0 , u1 )  f ( x1 , y1 , z1 , u1 )  Here

hu 

hx h y hz hu 16

1

1

1

1

 f ( x , y i

i 0 j 0 k 0 l 0

j

, z k , u l );

Dd . t

Considering (8.10), we present formulas for the approximate calculation of the n-fold integral on the trapezoidal:

Jn 

A1 A2 A3

An

   ...  f ( x1 , x2 ,..., xn )dx1dx2 dx3 ...dxn 

a1 a2 a3

an

1 2n

n

1

1

1

1

 h  ... f ( x i

i 1

i1 0 i2 0 i3 0

in 0

1i2

, x2i2 ,..., xnin ) (15)

Then for the formulas (6) and (8) - (11) we derive the general trapezoid formula. For this purpose, one-dimensional integral for integral intervals [a, b] is divided by n equal parts: [ x0 , x1 ],..., [ xn1 , xn ], xi  x0  ih x .

[a, b]  [c, d ] region of integration is divided, respectively, by n and m parts: [ x0 , x1 ],...,[ xn1 , xn ]; [ y0 , y1 ],...,[ y m1 , y m ] xi  x0  ih x ,

The two-dimensional integral of the

(16)

y j  y 0  jh y .

In the same way ,the three-, four-, etc. n-dimensional integration region is divided by the corresponding parts. In this approach, the formula (6) takes the form www.ijmer.com

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International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 b h n 1 h n 1 h n J 1   f ( x)dx   ( y i  y i 1 )   ( f i  f i 1 )   ( f i 1  f i )  2 i 0 2 i 0 2 i 1 (17) a n 1 f0  fn   fi . 2 i 0 The general trapezoid formula for double integrals calculating has the form

J2  

hx h y 4

hx h y 4

  f

n 1 m 1 i 0 j 0

n 1 m 1 1

 f i , j 1  f i 1, j  f i 1, j 1 

i, j

(18)

1

  f i  0 j  0 i1  0 i2  0

i  i1 j  i2

.

The following is a general formula for the trapezoidal three-and four-fold integrals calculating A B C

J3 



f ( x, y, z ) dxdydz 

hx h y hz 8

a b c

n 1 m 1 p 1

f i 0 j 0 k 0

 f i , j , k 1  f i , j 1, k 

i , j ,k

(19)

 f i , j 1, k 1  f i 1, j , k  f i 1, j , k 1  f i 1, j 1, k  f i 1, j 1, k  f i 1, j 1, k 1  

hx h y hz 8

n 1 m 1 p 1

1

1

1

 f i  0 j  0 k  0 i1  0 i2  0 i3  0

i  i1 , j  i2 , k  i3

ABCD

J 4      f ( x, y, z, u )dxdydzdu 

,

hx h y hz hu

a b c d

16

f

 f i , j ,k ,l 1  f i , j ,k 1,l 

i , j , k ,l

 f i , j ,k 1,l 1  f i , j 1,k ,l  f i , j 1,k ,l 1  f i , j 1,k 1,l  f i , j 1,k 1,l 1  f i 1, j ,k ,l  f i 1, j ,k ,l 

(20)

 f i 1, j ,k 1,l  f i 1, j ,k 1,l 1  f i 1, j 1,k ,l  f i 1, j 1,k ,l 1  f i 1, j 1,k 1,l   f i 1, j 1,k 1,l 1  

hx h y hz hu 16

n 1 m 1 p 1 t 1

1

1

1

1

 f i  0 j  0 k  0 l  0 i1  0 i2  0 i3  0 i4  0

i  i1 , j  i2 , k  i3 ,l  i4

.

Based on the above formula, the approximate calculation of n-fold integral, using the formula (D), takes the form

Jn 

1 2n

n

ni 1 n2 1 n3 1

nn 1 1

i 1

p1 0 p2 0 p3 0

pn 0i1 0 i2 0

1

1

 ( Ai  ai )    ...  ... f p1 i1 , p2 i2 ,...,pn in .

(21)

in  0

Thus the theorem is completely proved. Theorem 2. Approximation formulas for determining the values of the double integral, obtained by the repeated use of the trapezoid quadrature formula is calculated as follows:



f ( x, y )dxdy 

( D)

hx h y 4

n

m

  i 1 j 1

f ij ,

ij

1* 2 4 ... 2 1  2 4 4 ... 4 2    2 4 4 ... 4 2  . ij    ........................... 2 4 4 ... 4 2    1 2 2 ... 2 1

where

(22)

(23)

Proof. Let’s use the formula (12). Revealing this amount and similar terms, we obtain the following formula: hx h y 4

f n 1 m 1 i 0 j 0

ij

 f ij 1  f i 1, j  f i 1, j 1 

hx h y

 f11  2 f12  2 f13  ...  2 f1n 1  f1,n 4 2 f 21  4 f 22  4 f 23  ...  4 f 2, n 1  4 f 2 , n 

(24)

.............................................................. 4 f n 1,1  4 f n 1, 2  4 f n 1, 3  ...  4 f n 1, n 1  4 f n 1, n . f n ,1  2 f n , 2  f n ,3  ...  2 f n , n 1  f n , n 

The (17) can form the matrix elements - ij in the form (16) and the result (15), indicating the proof of the theorem. www.ijmer.com

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International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 Theorem 3. Approximate formula for calculating the value of triple integrals, obtained by the repeated use of the trapezoid quadrature formula, is calculated as follows:



f ( x, y, z )dxdydz 

( D)

hx h y hz 8

n

m

S

  i 1 j 1 k 1

ijk

f ijk ,

(25)

where

ijk

ijk

1 2 2 2 ... 2 1    2 4 4 4 ... 4 2   2 4 4 4 ... 4 2  ,   ..............................  2 4 4 4 ... 4 2    1 2 2 2 ... 2 1  

i  1; n,

 2 4 4 ... 4 4 2     4 8 8 ... 8 4 2    .............................. ,    4 8 8 ... 8 4 2   2 4 4 ... 4 4 2   

(26)

i  2, n  1.

Proof. Let’s use (13). Revealing this amount and similar terms, we obtain f 1,1,1  2 f 1,1, 2  2 f 1,1,3  2 f 1,1, 4  ...  2 f 1,1, n 1  f 1,1, n

2 f 1, 2,1  4 f 1, 2, 2  4 f 1, 2,3  4 f 1, 2, 4  ...  4 f 1,1,n 1  2 f 1, 2, n ....................................................................................... 2 f 1, n 1,1  4 f 1, n 1, 2  4 f 1, n 1,3  4 f 1, n 1, 4  ...  4 f 1, n 1, n 1  2 f 1, n 1, n f 1, n ,1  2 f 1, n , 2  2 f 1, n ,3  2 f 1, n , 4  ...  2 f 1, n , n 1  f 1, n , n 2 f 2,1,1  4 f 2,1, 2  4 f 2,1,3  4 f 2,1, 4  ...  4 f 2,1,n 1  2 f 2,1, n 4 f 2, 2,1  8 f 2, 2, 2  8 f 2, 2,3  8 f 2, 2, 4  ...  8 f 2, 2, n 1  4 f 2, 2,n ............................................................................................... 4 f 2,101 ,1  8 f 2,10n 1 , 2  8 f 2,10,3  8 f 2,10n 1 , 4  ...  8 f 2,10n 1 , n 1  4 f 2,10n 1 , n 2 f 2,11,1  4 f 2,11, 2  4 f 2,11,3  4 f 2,11, 4  ...  4 f 2,11, n 1  2 f 2,11,n .................................................................................................. 2 f n 1,1,1  4 f n 1,1, 2  4 f n 1,1,3  4 f n 1,1, 4  ...  4 f n 1,1, n 1  2 f n 1,1, n 4 f n 1, 2,1  8 f n 1, 2, 2  8 f n 1, 2,3  8 f n 1, 2, 4  ...  8 f n 1, 2, n 1  4 f n 1, 2, n ............................................................................................................ 4 f n 1, n 1,1  8 f n 1, n 1, 2  8 f n 1, n 1,3  8 f n 1,n 1, 4  ...  8 f n 1, n 1, n 1  4 f n 1,n 1,n 2 f nn, n ,1  4 f n 1, n , 2  4 f n 1, n ,n  4 f n 1, n ,n  ...  4 f n 1, n , n 1  2 f n 1, n , n f n ,1,1  2 f n ,1, 2  2 f n ,1,3  2 f n ,1, 4  ...  2 f n ,1, n 1  f n ,1, n

(27)

2 f n , 2,1  4 f n , 2, 2  4 f n , 2,3  4 f n , 2, 4  ...  4 f n , 2, n 1  2 f n , 2, n .............................................................................................................................. 2 f n , n 1,1  4 f n ,n 1, 2  4 f n , n 1,3  4 f n , n 1, 4  ...  4 f n , n 1, n 1  ...  2 f n , n 1, n f n , n ,1  2 f n ,n , 2  2 f n , n ,3  2 f n , n , 4  ...  4 f n , n , n 1  f n , n , n . From (19) we form the

ijk

elements, resulting in the relation (18), which shows the proof of the theorem.

Theorem 4. Approximation formulas for the values determining of the quadruple integrals, obtained by the repeated use of the trapezoid quadrature formula, is calculated as follows:

  ( D)

f ( x, y, z, u )dxdydzdu 

hx h y hz hu 16

n 1 m 1 l 1 s 1

  i 0 j 0 k 0 l 0

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ijkl

f ijkl ,

(28)

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www.ijmer.com  A11  A  21  .  An 1,1    An ,1

An1  Ann  A1n

An , j

2 4   A1, j  . 4  2

A12 A22

... ...

A1, n 1 A2 , n 1

. An 1, 2

... ... ...

. An 1, n 1

An , 2

1 2   . 2   1

4

4

8

8

. 8

. 8

4

4

International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645

An , n 1

2

2

2

...

4

4

4

...

4

. 4

. 4

. 4

... ...

. 4

2

2

2

...

2

2 ... 8 4  . ... . , ... 8 4   ... 4 1 

...

2

  ,  . An 1, n   Ann   1 2 , . 2  1  A1n A2 n

4

j  1, n  1

,

2 4 ... 4 2 4 4 8 ... 8 4  8    Ai1  Ain   . . ... . . , i  2, n  1, Aij   . 4 8 ... 8 4 8    2 4 ... 4 2 4 Proof. Disclose the amount (14) and obtain the following relations:

8

...

16

...

. 16

... ...

8

...

4 16 8  . . , 16 8   8 4 8

i  2,3,..., n  1, j  2,3,..., n  1

f 1111  2 f 1112  2 f 1113  2 f 1114  ...  2 f 111n 1  f 111,n

2 f 1121  4 f 1122  4 f 1123  4 f 1124  ...  4 f112,n 1  2 f 112,n 2 f 1131  4 f 1132  4 f 1133  4 f 1134  ...  4 f113,n 1  2 f 113,n ..................................................................................... 2 f 11,n 1,1  4 f 11,n 1, 2  4 f 11,n 1,3  4 f 11,n 1, 4  ...  4 f11,n 1,n 1  2 f 11,n 1,n f 11n1  2 f 11n 2  2 f 11n 3  2 f 11n 4  ...  2 f11n ,n 1  f 111,n

2 f1211  4 f 1212  4 f 1213  4 f 1214  ...  4 f121,n 1  2 f 121,n 4 f1221  8 f 1222  8 f1223  8 f 1224  ...  8 f 122,n 1  4 f 122,n 4 f1231  8 f 1232  8 f1233  8 f 1234  ...  8 f 123,n 1  4 f 123,n .......................................................................................... 4 f12n 1,1  8 f12n 1, 2  8 f12n 1,3  8 f 12n 1, 4  ...  8 f 12n 1,n 1  4 f 12n 1,n 2 f12n1  4 f 12n 2  4 f12n 3  4 f 12n 4  ...  4 f12n ,n 1  2 f 12n ,n ...................................................................................................... f 1n11  2 f 1n12  ...  2 f 1n 1,n 1  f 1n ,n

2 f 1n 21  4 f 1n 22  ...  4 f 1n 1, 2 n 1  2 f 1n , 2 n ........................................................... 2 f 1n ,n 1,1  4 f1nn1, 2  ...  4 f 1n ,n 1,n 1  2 f 1nn1,n f 1nn1  2 f 1nn2  ...  2 f 1nnn1  f 1nn,n f 2111  4 f 2112  4 f 2113  ...  4 f 211,n 1  2 f 211,n 4 f 2121  8 f 2122  8 f 2123  ...  8 f 212,n 1  4 f 212, n ......................................................................... 4 f 21n 1,1  8 f 21n 1, 2  8 f 21n 1,3  ...  8 f 21n 1, n 1  4 f 21n 1,n 2 f 21n1  4 f 21n 2  4 f 21n 3  ...  4 f 21n , n 1  2 f 21n , n

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International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645

4 f 2211  8 f 2212  8 f 2213  ...  8 f 221,n 1  4 f 221,n 8 f 2221  16 f 2222  16 f 2223  ...  16 f 222,n 1  8 f 222,n 8 f 2231  16 f 2232  16 f 2233  ...  16 f 223,n 1  8 f 223,n ......................................................................... 8 f 22n 1,1  16 f 22n 1, 2  16 f 22n 1,3  ...  16 f 22n 1,n 1  8 f 22n 1,n 4 f 22n1  8 f 22n 2  8 f 22n 3  ...  8 f 22n ,n 1  4 f 22n ,n ............................................................................................ 4 f n 1,n11  8 f n 1,n12  8 f n 1,n13  ...  8 f n 1,n ,1, n 1  4 f n 1,n ,1, n 8 f n 1,n 21  16 f n 1,n 22  16 f n 1,n 23  ...  16 f n 1,n , 2,n 1  8 f n 1,n , 2, n ......................................................................................................

2 f n ,n 1,11  4 f n ,n 1,12  4 f n ,n 1,13  ...  4 f n , n 1,1,n 1  2 f n ,n 1,1,n 4 f n ,n 1, 21  8 f n ,n 1, 22  8 f n , n 1, 23  ...  8 f n , n 1, 2,n 1  4 f n ,n 1, 2,n .............................................................................................. 4 f n ,n 1,n 1,1  8 f n ,n 1,n 1, 2  8 f n ,n 1,n 1,3  ...  8 f n ,n 1, n 1,n 1  4 f n ,n 1,n 1,n 2 f n ,n 1,n ,1  4 f n ,n 1,n , 2  4 f n ,n 1,n ,3  ...  4 f n ,n 1,n ,n 1  2 f n , n 1,n , n

8 f n 1,n,n 1,1  16 f n 1,n,n 1, 2  16 f n 1,n,n 1,3  ...  16 f n1,n,n 1,n 1  8 f n 1,n,n 1,n 4 f n 1,n,n,1  8 f n 1,n,n, 2  8 f n 1,n,n,3  ...  8 f n 1,n,n,n 1  8 f n 1,n,n,n f nn11  2 f nn12  2 f nn13  ...  2 f nn1,n 1  f nn1n 2 f nn21  4 f nn22  4 f nn23  ...  4 f nn2,n 1  4 f nn2,n ............................................................................ 2 f nnn1,1  4 f nnn1, 2  4 f nnn1,3  ...  4 f nn,n 1, n 1  4 f nn,n 1,n f nnn1  2 f nnn2  2 f nnn3  ...  2 f nnnn1  f nnn,n From these relations the formula (20) is formed, which fully demonstrates the proof of the theorem. Now, we calculate the values of integrals by repeated application of quadrature formulas, based on the formula of rectangles. The rectangle formula to calculate the values of the integrals is quite simple. In fact, here  i is taken as

[ xi 1 , xi ] midpoints. For a uniform grid (hi  h, i  1, n) we obtained formulas of the form b

J1 

f ( x)dx 

a

where

f i 1 / 2

h  f ( xi  ), 2

ba n  f i 1 / 2 , n i 1

(29)

x 0  a, x n  b .

Double integral in this case is calculated by the formula A B

J2 



f ( x, y )dxdy 

a b A

a

B b

 

n2

 f x, y

A

B

a

b

 dx  f ( x, y)dy 

n2

j 1

j 1 / 2

dx  

A  a B  b n1 n2  f xi 1 / 2 , y j 1 / 2   n1 n2 i 1 j 1

(30)

Aa Bb  f i 1 / 2, j 1 / 2 , n1 n2 i 1 j 1 n1

n2

hy   h Aa Bb f i 1 / 2, j 1 / 2  f  xi  x y j  , n1  , n2  ; n 2 h h x y   h x - Step on the OX axis, h y y - a step on the OY axis.

where

In the same way we define cubature formulas for calculating the values of the three-and four-fold integrals www.ijmer.com

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International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645 Aa B b C c n n n f ( x, y, z )dxdydz   f i 1 / 2, j 1 / 2,k 1 / 2 , (31) n1 n2 n3 i 1 j 1 k 1 www.ijmer.com

A BC



J3 

a b c

1

2

3

hy  hx hz f i 1 / 2, j 1 / 2,k 1 / 2  f   xi  2 , y j  2 , z k  2 

Where ABC D



J4 

f ( x, y, z, u )dxdydzdu 

a b c d

,   

Aa B b C c Dd n1 n2 n3 n4

n1

n2

n3

n4

 f i 1 j 1 k 1 l 1

i 1 / 2 , j 1 / 2 , k 1 / 2 ,l 1 / 2

; (32)

hy  h h  h f i 1 / 2, j 1 / 2,k 1 / 2,l 1 / 2  f  xi  x , y j  , z k  z , u l  u  . 2 2 2 2  

Here

On the basis of approximate formulas for calculating the values of one-, two-, three-and four-fold integrals, respectively, expressed by (21) - (24), the formula for calculating the values of n-fold integrals is given . Theorem 5. Let the y  f ( x1 , x2 ,..., xn ) function is defined and continuous in a n-dimensional bounded n integration domain. Then the cubature formula, obtained by repeated application of the rectangles’ formula, has the form nn n Ai  ai n1 n2 ... f i1 1 / 2,i2 1 / 2,...,in 1 / 2 , (33)     . ..D  f ( x1 , x2 ,..., xn )dx1dx2 ...dxn   ni i1 1 i2 1 in 1 i 1

xi  [ai , Ai ],

where

i  1, n ,

h  h h  f i1 1 / 2,i2 1 / 2,...,in 1 / 2  f  xi1  1 , xi2  2 ,..., xin  n  . 2 2 2   Proof. The proof is given by induction. From (21) - (24) we see that (25) holds for n  1;2;3;4 . We assume that (25) is valid for n

  . ..  f ( x , x ,..., x 1

D

2

k

k

)dx1dx2 ...dxk   i 1

k:

Ai  ai  ... f i 1/ 2,i2 1/ 2,...,ik 1 / 2 . k i i1 1 i2 1 ik 1 1 k1

k2

kk

Now we show fairness at n  k  1 :

  ...  f ( x , x 1

( D)

A

k

 a i 1

 i 1

,..., x k , x k 1 )dx1 dx 2 ...dx k dx k 1 

Ai  ai n1 n2 nk  h h h  ... f  xi1  1 , xi2  2 ,..., xik  k , x k 1 dx k 1   ni i1 1 i2 1 ik 1  2 2 2 

Ak 1  a k 1 nk 1 k 1

2

k

 i 1

Ai  ai n1 n2 nk nk 1  h h  h h  ...  f  xi  21 , xi2  22 ,..., xik  2k , xk 1  k21   ni i1 1 i2 1 ik 1 ik 1 1  1

n k n k 1 n2 Ai  ai ...  f i1 1 / 2,i2 1 / 2,...,ik 1 / 2,ik 1 1 / 2 .  ni i1 1 i2 1 ik 1 ik 1 1 n1

The obtained result proofs the theory completely.

III. CONCLUSIONS Thus we have constructed the multi-dimensional cubature formulas to approximate the value of multiple integrals by repeated application of the quadrature formula of trapezoids and rectangles to determine the values of multiple integrals, which are easy to implement on a computer.

REFERENCES [1]. [2]. [3]. [4].

[5]. [6]. [7]. [8]. [9]. [10].

Demidovich B.L, Maron I.A. “Foundations of Computational Mathematics”. - Moscow: Nauka, 1960. - 664 p. Rvachev V.L. “Theory of R-functions and some of its applications”. - Naukova. Dumka, 1982. - 552 p. Rvachev V.L. “An analytical description of some geometric objects”. Dokl. Kiev, 1963. - 153, № 4. 765-767pp. Nazirov S.A. “Constructive definition of algorithmic tools for solving three-dimensional problems of mathematical physics in the areas of complex configuration” / Iss. Comput. and glue. Mathematics. Tashkent .Scientific works set. IMIT Uzbek Academy of Sciences, 2009. - No. 121. - 15-44pp. Kabulov V.K., Faizullaev A., Nazirov S.A. “Al-Khorazmiy algorithm, algorithmization”. - Tashkent: FAN, 2006. -664 p. Gmurman V.E. “Guide to solving problems in the theory of probability and mathematical statistics”. Manual. book for students of technical colleges. - 3rd ed., Rev. and add. - M.: Higher. School, 1979. Ermakov S.M. “Monte Carlo methods and related issues”. - Moscow: Nauka, 1971. Sevastyanov B.A. “The course in the theory of probability and mathematical statistics”. - Moscow: Nauka, 1982. Mathematics. Great Encyclopedic Dictionary/ Ch. Ed. V. Prokhorov. - Moscow: Great Russian Encyclopedia, 1999. Gmurman E. “Probability theory and mathematical statistics.” Textbook for technical colleges. 5th ed., revised. and add. - M., Higher. School, 1977.

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