The Method of Repeated Application of a Quadrature Formula of Trapezoids and Rectangles to Determine the Values of Multiple Integrals

International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol. 3, Issue. 4, Jul - Aug. 2013 pp-2027-2034 ISSN: 2249-6645
www.ijmer.com 2027 | Page
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]
))()(()(),( xxxyxy  
and two vertical lines bxax  , (Fig.1).
Placing the known rules in the double integral

)(
),(
D
dxdyyxfI (1)
limits of integration, we obtain
 
)(
)(
)(
.),(),(
D
x
x
b
a
dyyxfdxdxdyyxf


Let

)(
)(
.),()(
x
x
dyyxfxF


(2)
Then
.)(),(
)(
 
D
b
a
dxxFdxdyyxf (3)
Fig.1
Applying to a single integral in the right-hand side of (3), one of the quadrature formulas, we have
 

)( 1
),(),(

n
i
ii xFAdxdyyxf (4)
where - ),...,2,1(],[ nibaxi  and iA are some constant coefficients. In turn, the value
y
x0 а b
)(xy 
)(xy 

The Method of Repeated Application of a Quadrature Formula of
Trapezoids and Rectangles to Determine the Values of Multiple Integrals


)(
)(
),()(
i
i
x
x
ii dyyxfxF


can also be found on some quadrature formulas
,),()(
1


im
j
jiiji yxfBxF
where ijB iy - appropriate constants.
From (4) we obtain
  

)( 1 1
),,(),(
D
n
i
m
j
jiiji
i
yxfBAdxdyyxf (5)
where iA and ijB iy - are known constants.
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
   

n
i
m
j
kji
t
k
kijiji
D
i i
zyxfCBAdxdydzzyxf
1 1 1
,
)(
),,(),,( .
Now these results are generalized to compute the values of multiple integrals
),,...,,(......
...),,(...
21
)()3()2(
1 1 1
)1(
2121
)(
21321321
1
1
2
2
1 iiik
k
n
k
k
iiiiiiiii
n
i
n
i
n
i
i
kk
D
xxxfAAAA
dxdxdxxxxf
 

  


Where
)(
...
)2()1(
21211
,...,, k
iiiiii k
AAA - 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
),...,,( 21 nxxxfy 
Is continuous in a bounded domain D. Then the formula for the approximate values of the n-fold integral calculating
  nn
D
n dxdxdxxxfJ ,...,),...,(... 211
)(
, (6) based on the trapezoidal
rule has the form
   

   








1
0
1
0
1
0
1
0
,...,,
1
0 0
1
01 1 2
2211
1
1
2
2
3
3
......
n
n n
nn
n
p i i i
ipipip
n
p
n
p
n
pi
in fhJ
Here D is n-dimensional domain of integration of the form
niAxa iii ,1,  ; (7)
2
ii
i
aA
h


and it is believed that each interval (5), is respectively, divided into nnnn ,...,, 21 parts.
Proof. Approximate formula for calculating the values of the one-dimensional integral in this case is
   

1
0
101
2
)()(
2
)(
1
0
i
i
x
x
x
x
f
h
xfxf
h
dxxfJ , (8)
where,
n
ab
hx

 , n - the number of the interval divisions into n parts. In (4) n = 1 .
Now we define the approximate calculation formula for double integrals

)(
2 ),(
D
dxdyyxfJ (9)

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,
bxa  , dyc  .
Then we rewrite the integral (7) in the form

d
c
b
a
dyyxfdxJ ),(2 . (10)
Using the trapezoidal rule to calculate the inner integral, we obtain






 
b
a
b
a
y
dxyxfdxyxf
h
J ),(),(
2
2 . (11)
Now to each integral we use the trapezoid formulas and the result is
   
  ,),(
4
),(),(),(),(
4
),(),(
2
),(),(
22
1
0
1
0
11100100
111001002
 









i j
ji
yxyx
xxy
yxf
hh
yxfyxfyxfyxf
hh
yxfyxf
h
yxfyxf
hh
J
(12)
where
m
cd
hy

 .
Similarly, for the approximate calculation of the values of the triple integral we obtain the following formula:


,),,(
8
),,(),(),(),,(),,(),,(
),,(),,(),,(
8
),,(
1
0
1
0
1
0
111011011101001110
0101000003


  


i j k
kji
zyx
zyx
A
a
B
b
C
c
zyxf
hhh
zyxfzyxfzyxfzyxfzyxfzyxf
zyxfzyxfzyxf
hhh
dxdydzzyxfJ
(13)
where
p
cC
hz

 .
The four-dimensional integral is defined with 16 members:

 ;),,,(
16
),,,(),,,(
),,,(),,,(),,,(),,,(),,,(
),,,(),,,(),,,(),,,(),,,(
),,,(),,,(),,,(
16
),,,(
1
0
1
0
1
0
1
0
11111011
00111101010110010001
11100110101000101100
0100100000004


   




i j k l
lkji
uzyx
uzyx
A
a
B
b
C
c
D
d
uzyxf
hhhh
uzyxfuzyxf
uzyxfuzyxfuzyxfuzyxfuzyxf
uzyxfuzyxfuzyxfuzyxfuzyxf
uzyxfuzyxfuzyxf
hhhh
dxdydzduuzyxfJ
(14)
Here
t
dD
hu

 .
Considering (8.10), we present formulas for the approximate calculation of the n-fold integral on the trapezoidal:
        

1
0
1
0
1
0
1
0
21
1
32121
1 2 3
22
1
1
2
2
3
3
),...,,(...
2
1
...),...,,(...
i i i i
niii
n
i
in
A
a
A
a
A
a
A
a
nnn
n
n
n
n
xxxfhdxdxdxdxxxxfJ (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: xinn ihxxxxxx  0110 ],,[],...,,[ .
The two-dimensional integral of the ],[],[ dcba  region of integration is divided, respectively, by n and m parts:
],[],...,,[];,[],...,,[ 110110 mmnn yyyyxxxx 
.
,
0
0
yj
xi
jhyy
ihxx


(16)
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

.
2
)(
2
)(
2
)(
2
)(
1
0
0
1
1
1
0
1
1
0
11
















n
i
i
n
n
i
ii
n
i
ii
n
i
ii
b
a
f
ff
ff
h
ff
h
yy
h
dxxfJ
(17)
The general trapezoid formula for double integrals calculating has the form
 
.
4
4
1
0
1
0
1
0
1
0
1
0
1
0
1,1,11,,2
1 2
21




  








n
i
m
j i i
ijii
yx
n
i
m
j
jijijiji
yx
f
hh
ffff
hh
J
(18)
The following is a general formula for the trapezoidal three-and four-fold integrals calculating


,
8
8
),,(
1
0
1
0
1
0
1
0
1
0
1
0
,,
1,1,1,1,1,1,11,,1,,11,1,
1
0
1
0
1
0
,1,1,,,,3
1 2 3
321
  





   












n
i
m
j
p
k i i i
ikijii
zyx
kjikjikjikjikjikji
n
i
m
j
p
k
kjikjikji
A
a
zyx
B
b
C
c
f
hhh
ffffff
fff
hhh
dxdydzzyxfJ
(19)


.
16
16
),,,(
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
,,,
1,1,1,1,1,1,11,,1,1,,1,11,1,,1,1,,1
,,,1,,,11,1,1,,1,1,1,,1,,,1,1,1,,
,1,,1,,,,,,4
1 2 3 4
4321








    








n
i
m
j
p
k
t
l i i i i
ilikijii
uzyx
lkjilkjilkjilkjilkjilkji
lkjilkjilkjilkjilkjilkjilkji
lkjilkjilkji
A
a
uzyx
B
b
C
c
D
d
f
hhhh
ffffff
fffffff
fff
hhhh
dudxdydzuzyxfJ
(20)
Based on the above formula, the approximate calculation of n-fold integral, using the formula (D), takes the form
.......)(
2
1 1
0
1
0
1
0
1
0
1
0
1
0
1
0
,...,,
1 1
2
2
3
3 1 2
2211   







   



i n
n n
nn
n
p
n
p
n
p
n
p i i i
ipipip
n
i
iinn faAJ (21)
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:
  

n
i
m
j
ijij
D
yx
f
hh
dxdyyxf
1 1)(
4
),(  , (22)
where





















12...221
24...442
...........................
24...442
24...442
12...421*
ij . (23)
Proof. Let’s use the formula (12). Revealing this amount and similar terms, we obtain the following formula:
 

nnnnnnn
nnnnnnn
nn
nn
yx
n
i
m
j
jijiijij
yx
fffff
fffff
fffff
fffff
hh
ffff
hh
,1,3,2,1,
,11,13,12,11,1
,21,2232221
,111131211
1
0
1
0
1,1,11
2...2
.44...444
..............................................................
44...442
2...22
4
4














(24)
The (17) can form the matrix elements - ij in the form (16) and the result (15), indicating the proof of the theorem.

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:
   

n
i
m
j
S
k
ijkijk
zyx
D
f
hhh
dxdydzzyxf
1 1 1)(
8
),,(  , (25)
where
.1,2,
244...442
248...884
..............................
248...884
244...442
,;1,
12...2221
24...4442
..............................
24...4442
24...4442
12...2221








































ni
ni
ijk
ijk


(26)
Proof. Let’s use (13). Revealing this amount and similar terms, we obtain
.4...222
2...4...4442
..............................................................................................................................
24...4442
2...222
24...4442
48...8884
............................................................................................................
48...8884
24...4442
..................................................................................................
24...4442
48...8884
...............................................................................................
48...8884
24...4442
2...222
24...4442
.......................................................................................
24...4442
2...222
,,1,,4,,3,,2,,1,,
,1,1,1,4,1,3,1,2,1,1,1,
,2,1,2,4,2,3,2,2,2,1,2,
,1,1,1,4,1,3,1,2,1,1,1,
,,11,,1,,1,,12,,11,,
,1,11,1,14,1,13,1,12,1,11,1,1
,2,11,2,14,2,13,2,12,2,11,2,1
,1,11,1,14,1,13,1,12,1,11,1,1
,11,21,11,24,11,23,11,22,11,21,11,2
,10,21,10,24,10,23,10,22,10,21,10,2
,2,21,2,24,2,23,2,22,2,21,2,2
,1,21,1,24,1,23,1,22,1,21,1,2
,,11,,14,,13,,12,,11,,1
,1,11,1,14,1,13,1,12,1,11,1,1
,2,11,1,14,2,13,2,12,2,11,2,1
,1,11,1,14,1,13,1,12,1,11,1,1
11111
nnnnnnnnnnnnnn
nnnnnnnnnnnnnn
nnnnnnnn
nnnnnnnn
nnnnnnnnnnnnnnnnn
nnnnnnnnnnnnnn
nnnnnnnn
nnnnnnnn
nn
nn
nn
nn
nnnnnnnn
nnnnnnnn
nn
nn
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
nnnn

































(27)
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:
 









1
0
1
0
1
0
1
0)(
16
),,,(
n
i
m
j
l
k
s
l
ijklijkl
uzyx
D
f
hhhh
dxdydzduuzyxf  , (28)























nn
nn
nn
nn
n
n
n
n
nn
nn
A
A
A
A
A
A
A
A
AAAA
AAAA
,1
1,
1,1
2,
2,1
1,
1,1
21,22221
11,11211
...
...
.......
...
...
,

















1
2
.
2
1
2...2
4...4
2
4
2
4
1
2
........
4...4442
2...2221
11 nnnn AAA
,
1,1,
14...
48...
4
8
4
8
2
4
........
48...884
24...442
,1, 
















 njAA jjn
,
1,...,3,2
,1,...,3,2
,
4
8
8
16
...
...
8
16
4
8
.......
816...168
48...84
,1,2,
2
4
4
8
...
...
4
8
2
4
.......
48...84
24...42
1




































nj
ni
AniAA ijini
Proof. Disclose the amount (14) and obtain the following relations:
nnnnnnn
nnnnnnnn
nn
nn
nn
ffffff
ffffff
ffffff
ffffff
ffffff
,1111,11411311211111
,1,111,1,114,1,113,1,112,1,111,1,11
,1131,1131134113311321131
,1121,1121124112311221121
,11111111114111311121111
2...222
24...4442
.....................................................................................
24...4442
24...4442
2...222










......................................................................................................
24...4442
48...8884
..........................................................................................
48...8884
48...8884
24...4442
,121,12412312212112
,1121,1124,1123,1122,1121,112
,1231,1231234123312321231
,1221,1221224122312221221
,1211,1211214121312121211
nnnnnnnn
nnnnnnnn
nn
nn
nn
ffffff
ffffff
ffffff
ffffff
ffffff










nnnnnnnnnn
nnnnnnnnnn
nnnnnn
nnnnnn
ffff
ffff
ffff
ffff
,1112111
,111,1,12,111,1,1
2,112,11221211
,11,11121111
2...2
24...42
...........................................................
24...42
2...2








nnnnnnn
nnnnnnn
nn
nn
fffff
fffff
fffff
fffff
,211,21321221121
,1211,1213,1212,1211,121
,2121,212212321222121
,2111,211211321122111
24...442
48...884
.........................................................................
48...884
24...44









............................................................................................
48...884
816...16168
.........................................................................
816...16168
816...16168
48...884
,221,22322222122
,1221,1223,1222,1221,122
,2231,223223322322231
,2221,222222322222221
,2211,221221322122211
nnnnnnn
nnnnnnn
nn
nn
nn
fffff
fffff
fffff
fffff
fffff










......................................................................................................
816...16168
48...884
,2,,11,2,,123,122,121,1
,1,,11,1,,113,112,111,1
nnnnnnnnnnnn
nnnnnnnnnnnn
fffff
fffff




nnnnnnnnnnnnnnnnn
nnnnnnnnnnnnnnnnn
nnnnnnnnnnnn
nnnnnnnnnnnn
fffff
fffff
fffff
fffff
,,1,1,,1,3,,1,2,,1,1,,1,
,1,1,1,1,1,3,1,1,2,1,1,1,1,1,
,2,1,1,2,1,23,1,22,1,21,1,
,1,1,1,1,1,13,1,12,1,11,1,
24...442
48...884
..............................................................................................
48...884
24...442








nnnnnnnnnnnnnnnnn
nnnnnnnnnnnnnnnnn
fffff
fffff
,,,11,,,13,,,12,,,11,,,1
,1,,11,1,,13,1,,12,1,,11,1,,1
88...884
816...16168




nnnnnnnnnnnnnnnnn
nnnnnnnnnnnnnnnnn
nnnnnnnnnnnn
nnnnnnnnnnnn
fffff
fffff
fffff
fffff
,1321
,1,1,1,3,12,11,1
,21,2232221
11,1131211
2...22
44...442
............................................................................
44...442
2...22








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
],[ 1 ii xx  midpoints. For a uniform grid ),1,( nihhi  we obtained formulas of the form
 



n
i
i
b
a
f
n
ab
dxxfJ
1
2/11 )( , (29)
where bxax
h
xff nii  ,),
2
( 02/1 .
Double integral in this case is calculated by the formula
   
,
,,
),(),(
1 2
1 22
1 1
2/1,2/1
21
1 1
2/12/1
211
2/1
2
2

 

 

 












 


n
i
n
j
ji
n
i
n
j
ji
A
a
n
j
j
B
b
A
a
A
a
B
b
f
n
bB
n
aA
yxf
n
bB
n
aA
dxyxf
n
bB
dyyxfdxdxdyyxfJ
(30)
where ;,,
2
212/1,2/1
yx
y
j
x
iji
h
bB
n
h
aA
n
h
y
n
h
xff










xh - Step on the OX axis, yh y - a step on the OY axis.
In the same way we define cubature formulas for calculating the values of the three-and four-fold integrals

   



1 2 3
1 1 1
2/1,2/1,2/1
321
3 ),,(
n
i
n
j
n
k
kji
A
a
B
b
C
c
f
n
cC
n
bB
n
aA
dxdydzzyxfJ , (31)
Where







2
,
2
,
2
2/1,2/1,2/1
z
k
y
j
x
ikji
h
z
h
y
h
xff ,
    



1 2 3 4
1 1 1 1
2/1,2/1,2/1,2/1
4321
4 ),,,(
n
i
n
j
n
k
n
l
lkji
A
a
B
b
C
c
D
d
f
n
dD
n
cC
n
bB
n
aA
dxdydzduuzyxfJ ; (32)
Here







2
,
2
,
2
,
2
2/1,2/1,2/1,2/1
u
l
z
k
y
j
x
ilkji
h
u
h
z
h
y
h
xff .
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 ),...,,( 21 nxxxfy  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
     




1
1
2
2
21
1 1 1
2/1,...,2/1,2/1
1
2121 ......),...,,(...
n
i
n
i
n
i
iii
n
i i
ii
nn
D
n
n
n
f
n
aA
dxdxdxxxxf , (33)
where niAax iii ,1],,[  ,







2
,...,
2
,
2
21
2/1,...,2/1,2/1 2121
n
iiiiii
h
x
h
x
h
xff nn
.
Proof. The proof is given by induction.
From (21) - (24) we see that (25) holds for 4;3;2;1n . We assume that (25) is valid for kn  :
     




1
1
2
2
21
1 1 1
2/1,...,2/1,2/1
1
2121 ......),...,,(...
k
i
k
i
k
i
iii
k
i i
ii
kk
D
k
k
k
f
k
aA
dxdxdxxxxf .
Now we show fairness at 1 kn :
....
2
,
2
,...,
2
,
2
...
,
2
,...,
2
,
2
...
...),,...,,(...
1
1
2
2
1
1
121
21
1
1
2
2
1
1
1
1
2
2
21
1 1 1 1
2/1,2/1,...,2/1,2/1
1
1
1
1
21
1 1 1 111
11
1 1 1
11
21
1
121121
)(
  
  
 
 
   





   

  





























n
i
n
i
n
i
n
i
iiii
k
i i
ii
k
k
k
iii
n
i
n
i
n
i
n
i
k
i i
ii
k
kk
n
i
n
i
n
i
kk
k
iii
A
a
k
i i
ii
kkkk
D
k
k
k
k
kk
k
k
k
k
k
k
k
k
f
n
aA
h
x
h
x
h
x
h
xf
n
aA
n
aA
dxx
h
x
h
x
h
xf
n
aA
dxdxdxdxxxxxf
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]. Demidovich B.L, Maron I.A. “Foundations of Computational Mathematics”. - Moscow: Nauka, 1960. - 664 p.
[2]. Rvachev V.L. “Theory of R-functions and some of its applications”. - Naukova. Dumka, 1982. - 552 p.
[3]. Rvachev V.L. “An analytical description of some geometric objects”. Dokl. Kiev, 1963. - 153, № 4. 765-767pp.
[4]. 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.
[5]. Kabulov V.K., Faizullaev A., Nazirov S.A. “Al-Khorazmiy algorithm, algorithmization”. - Tashkent: FAN, 2006. -664 p.
[6]. 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.
[7]. Ermakov S.M. “Monte Carlo methods and related issues”. - Moscow: Nauka, 1971.
[8]. Sevastyanov B.A. “The course in the theory of probability and mathematical statistics”. - Moscow: Nauka, 1982.
[9]. Mathematics. Great Encyclopedic Dictionary/ Ch. Ed. V. Prokhorov. - Moscow: Great Russian Encyclopedia, 1999.
[10]. Gmurman E. “Probability theory and mathematical statistics.” Textbook for technical colleges. 5th ed., revised. and add. - M.,
Higher. School, 1977.

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