CONTRIBUTIONS TO THE NUMERICAL INTEGRATION OF FUNCTIONS OF MULTIPLE VARIABLES

Communication presented at the meeting of September 24, 1956 of the Cluj Branch of the RPR Academy

§ 1. General considerations

1.

Let me note $D$ a domain in Euclidean space $S$ - dimensional $E_{s}$ , with $you_{M}$ a volume element from $D$ , with $f(M)$ and $K(M)$ two point functions $M=M\left(t^{1},t^{2},\ldots,t^{S}\right)$ , integrable in the considered domain.

By numerical integration formula or cubature formula is meant a formula of the form

\iint\ldots\int_{D}K(M)f(M)dv_{M}=\sum_{i=1}^{N}c_{i}f\left(M_{i}\right)+\rho

(1)

where the numbers $c_{i}$ - which are called the coefficients of the formula - depend only on the nodes $M_{i}$ of the cubature formula, which are points in $D;\rho$ is the rest of this formula.

The finite sum of the second term

J(f)=\sum_{i=1}^{N}c_{i}f\left(M_{i}\right)

(2)

represents an approximate assessment of the functionality

I(f)=\iint\ldots\int_{D}K(M)f(M)dv_{M}

(3)

where the function $K(M)$ it is assumed that it is chosen once and for all for the function $I(f)$ and that it keeps a constant sign in $D$ .

Therefore, a cubature formula is a formula that allows an approximate evaluation of a definite integral of a function to be given. $f(M)$ , multiplied by a weight function $K(M)$ , through a certain linear combination of the values of the function $f(M)$ on a finite number of distinct points. We also have
cubature formulas in which the values of the partial derivatives of $f(M)$ at certain points.

REST $\rho$ of formula (1) represents the value, given by this formula, of an additive and homogeneous functional, which, to indicate the function $f(M)$ , we will also denote it by $\rho(f)$ .
2. We will say that formula (1) has a partial degree of accuracy ( $n_{1},n_{2},\ldots n_{s}$ ). if :
a) $\rho(P)=0$ for any polynomial $P(M)$ of degree $\left(n_{1},n_{2},\ldots,n_{s}\right)$ .
b) $\rho(P)\neq 0$ for the least polynomial $P(M)$ of degree $\left(n_{1}+1\right.$ , $\left.n_{2}+1,\ldots,n_{s}+1\right)$ .
3. The basic problems that now arise are the following:
a) To construct the cubature formulas, that is, to determine the coefficients $c_{i}$ and the nodes $M_{i}$ .
b) Give and study the expression of the rest of these formulas in order to be able to evaluate the error that is committed when for $I(f)$ the approximate value is taken $J(f)$ It
is known that a general method of constructing cubature formulas consists of replacing the function $f(M)$ by its expression given by an interpolation formula.

If we assume that $f(M)$ is expandable in the uniformly convergent series in $D$

f(M)=\sum_{i_{1},\ldots,i_{s}=0}^{\infty}a_{i_{1}\ldots i_{s}}\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}

(4)

and we put

I_{i_{1}\ldots i_{s}}=\iint\ldots\int_{D}K\left(t^{1},\ldots,t^{s}\right)\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}dv_{M}

the rest will be represented by the series

\rho=\sum_{i_{1},\ldots,i_{s}=0}^{\infty}a_{i_{1}\ldots i_{s}}\left[I_{i_{1}\ldots i_{s}}-\sum_{j=1}^{N}p_{j}\left(t_{j}^{1}\right)^{i_{1}}\ldots\left(t_{j}^{s}\right)^{i_{s}}\right],

(5)

the quantities in square brackets are all independent of $f(M)$ If
$f(M)$ is a polynomial of degree ( $n_{1},n_{2},\ldots,n_{s}$ ), we have

I(f)=J(f)

whatever the coefficients $a_{i_{1}}\ldots i_{s}$ for which $i_{k}\leqq n_{k}(k=1,s)$ This
leads us to the following equations

\sum_{j=1}^{N}p_{j}\left(t^{1}\right)^{i_{1}}\ldots\left(t^{s}\right)^{i_{s}}=I_{i_{1}\ldots i_{s}}

4.

It is natural to seek to increase the precision of the cubature formula we are talking about, seeking to determine the $sN$ unknown ( $t_{i}^{1},t_{i}^{2},\ldots,t_{i}^{s}$ ), $i=1,N$ so that it cancels out, whatever the function is $f(M)$ , and those $sN$ following terms in the series (5), deducing in this case $\bmod sN$ new equations of the form

\sum_{j=1}^{N}p_{j}\left(t_{j}^{1}\right)^{i_{1}}\ldots\left(t_{j}^{s}\right)^{i_{s}}=I_{i_{1}\ldots i_{s}}

In this way, a total system of $(s+1)N$ equations with the same number of unknowns

c_{i};t_{i}^{1},t_{i}^{2},\ldots,t_{i}^{s}\quad(i=\overline{1,N})

For the problem to be possible, the resulting linear system must be compatible and lead us to $N$ puncture $M_{i}$ real belonging to the field $D$ and without being located on a hypersurface of the order $\left(n_{1},n_{2},\ldots,n_{s}\right)$ in order not to cancel certain determinants that will intervene in the denominators of the coefficient expressions $c_{\mathrm{i}}$ 5.
In a previous work [3], relative to the system of $N=\left(n_{1}+1\right)\ldots\ldots\left(n_{s}+1\right)$ knots

M_{i_{1}\ldots i_{s}}=M_{i_{1}\ldots i_{s}}\left(t_{i_{1}}^{1},t_{i_{1}i_{2}}^{2},\ldots,t_{i_{1}\ldots i_{s}}^{s}\right)

(6)

I gave the interpolation formula

f(M)=L_{n_{1}n_{2}\ldots n_{s}}(M)+R_{s}(M)

(7)

where

L_{n_{1}\ldots n_{s}}(M)=\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{s}=1}^{n_{s}+1}l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{s}}^{s}\left(t^{s}\right)f\left(M_{i_{1}\ldots i_{s}}\right)

(8)

with

\begin{gathered}l_{i_{1}\ldots i_{k}}^{k}\left(t^{k}\right)=\frac{u_{i_{1}\ldots i_{k-1}}^{k}\left(t^{k}\right)}{\left(t^{k}-t_{i_{1}\ldots i_{k}}^{k}\right)\dot{u}_{i_{1}\ldots i_{k-1}}^{k}\left(t_{i_{1}\ldots i_{k}}^{k}\right)}\\ u_{i_{1}\ldots i_{k-1}}^{k}\left(t^{k}\right)=\prod_{i_{k}=1}^{n_{k}+1}\left(t^{k}-t_{i_{1}\ldots i_{k}}^{k}\right)\end{gathered}

is the interpolation polynomial of degree ( $n_{1},n_{2},\ldots,n_{s}$ ) which coincides with $f(M)$ on nodes (6):

The rest of formula (7) has the expression

R_{s}(M)=\sum_{p=1}^{s}\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{p-1}=1}^{n_{p-1}+1}l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{p-1}}^{p-1}\left(t^{p-1}\right)u_{i_{1}\ldots i_{p-1}}^{p}S_{i_{1}\ldots i_{p-1}}^{p},

where

S_{i_{1}\ldots i_{p-1}}^{p}=\left[t^{p},t_{i_{1}\ldots i_{p-1}1}^{p},\ldots,t_{i_{1}\ldots i_{p-1}}^{p},n_{p+1};f\left(t_{i_{1}}^{1},\ldots,t_{i_{1}\ldots i_{p-1}}^{p-1},t^{p},\ldots,t^{s}\right)\right]

Here on the right-hand side we have the difference divided by the nodes $t^{p},t_{i_{1}i_{2}\ldots i_{p-1}}^{p},\ldots\ldots,t_{i_{1}i_{2}\ldots i_{p-1}}n_{p}+1$ applied to the variable $t^{p}$ of the function

f\left(t_{i_{1}}^{1},t_{i_{1}i_{2}}^{2},\ldots,t_{i_{1}\ldots i_{p-1}}^{p-1},t^{p},t^{p+1},\ldots,t^{s}\right).

6.

Suppose that the points (6) belong to the domain $D$ If formula (7) is used, the cubature formula is obtained

\iint\ldots\int_{D}K(M)f(M)dv_{M}=\sum_{i_{1}=1}^{n_{1}+1}\ldots\sum_{i_{\mathrm{s}}=1}^{n_{s}+1}A_{i_{1}\ldots i_{s}}t\left(M_{i_{1}\ldots i_{s}}\right)+\rho_{s}

(9)

where the coefficients are given by the formula

A_{i_{1}\ldots i_{s}}=\iint\ldots\int_{D}K\left(t^{1},\ldots,t^{s}\right)l_{i_{1}}^{1}\left(t^{1}\right)\ldots l_{i_{1}\ldots i_{s}}^{s}\left(t^{s}\right)dt^{1}\ldots dt^{s}

(10)

and the rest is given by

\rho_{s}=\iint\cdots\int_{D}K(M)R_{s}(M)dM

(11)

7.

Example. Considering the case of $s=2$ gnarly

	$\displaystyle M_{1}(-a,-b),M_{2}\left(-a,-\frac{b-c}{2}\right),M_{3}(-a,c)$
	$\displaystyle M_{4}\left(0,-\frac{b+c}{2}\right),M_{6}(0,0),M_{6}\left(0,\frac{b+c}{2}\right)$		(12)
	$\displaystyle M_{7}(a,-c),M_{8}\left(a,\frac{b-c}{2}\right),M_{9}(a,b)$

formula (7) leads us to the interpolation formula

		$\displaystyle f(x,y)=\frac{1}{a^{2}(b+c)^{2}}x(x-a)\left(y+\frac{b-c}{2}\right)(y+b)f(-a,c)-$
		$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}x(x-a)(y+b)(y-c)f\left(-a,\frac{c-b}{2}\right)+$
		$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x-a)(y-c)\left(y+\frac{b-c}{2}\right)f(-a,-b)-$
		$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)y\left(y+\frac{b+c}{2}\right)f\left(0,\frac{b+c}{2}\right)+$

	$\displaystyle+\frac{4}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)\left[y^{2}-\frac{(b+c)^{2}}{4}\right]f(0,0)-$
	$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}\left(x^{2}-a^{2}\right)\left(y-\frac{b+c}{2}\right)yf\left(0,-\frac{b+c}{2}\right)+$		(13)
	$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x+a)(y+c)\left(y-\frac{b-c}{2}\right)f(a,b)-$
	$\displaystyle-\frac{2}{a^{2}(b+c)^{2}}x(x+a)(y-b)(y+c)f\left(a,\frac{b-c}{2}\right)+$
	$\displaystyle+\frac{1}{a^{2}(b+c)^{2}}x(x+a)(y-b)\left(y-\frac{b-c}{2}\right)f(a,-c)+r(x,y)$

where the remainder has the expression

	$\displaystyle r(x,y)=x\left(x^{2}-a^{2}\right)[-a,0,a,x;f(x,y)]+$
	$\displaystyle+\frac{1}{2a^{2}}x(x-a)(y+b)(y-c)\left(y+\frac{b-c}{2}\right)\cdot\left[c,\frac{c-b}{2},-b,y;f(a,y)\right]-$
	$\displaystyle-\frac{1}{a^{2}}\left(x^{2}-a^{2}\right)y\left(y^{2}-\frac{(b+c)^{2}}{4}\right)\left[\frac{b+c}{2},0,-\frac{b+c}{2},y;f(0,y)\right]+$
	$\displaystyle+\frac{1}{2a^{2}}x(x+a)(y-b)\left(y-\frac{b-c}{2}\right)(y+c)\left[b,\frac{b-c}{2},-c,y;f(-a,y)\right]$		(14)

Taking the parallelogram as the domain of integration $D$ of peaks

M_{1}(-a,-b),M_{3}(-a,c),M_{7}(a,-c),M_{9}(a,b)

(15)

and doing $K(x,y)=1$ , the cubature formula (9) becomes

\displaystyle\begin{array}[]{l}\iint_{D}f(x,y)dxdy=\frac{a}{90(b+c)}\left\{\left(22bc-b^{2}-c^{2}\right)[f(a,b)+f(-a,c)+\right.\\ +f(-a,-b)+f(a,-c)]+8\left(7b^{2}+26bc+7c^{2}\right)f(0,0)+\\ +16\left(2b^{2}+bc+2c^{2}\right)\left[f\left(0,\frac{b+c}{2}\right)+f\left(-a,\frac{c-b}{2}\right)+\right.\\ \left.\left.+f\left(0,-\frac{b+c}{2}\right)+f\left(a,\frac{b-c}{2}\right)\right]\right\}+\rho^{\prime}\end{array}

\mathrm{p}^{\prime}=\iint_{D}r(x,y)dxdy

(17)

The cubature formula (16) has a partial degree of accuracy ( 2.2 ) and a global degree of accuracy equal to 3.

If we do $d=c$ we obtain the following partial-accuracy cubature formula $(3,3)$

	$\displaystyle\iint_{D}f(x,y)dxdy=$
	$\displaystyle=\frac{ab}{9}\{f(a,b)+f(-a,b)+f(-a,-b)+f(a,-b)+4[f(0,b)+$
	$\displaystyle+f(-a,0)+f(0,-b)+f(a,0)]+16f(0,0)\}+\rho$		(18)

which is precisely the classical Cavalieri-Simpson formula extended to two variables. The domain of integration $D$ is in this case the rectangle defined by the inequalities

-a\leqq x\leqq a,\quad-b\leqq y\leqq b

(19)

The rest of the Cavalieri-Simpson cubature formula is given by the formula

\rho=\iint_{D}R(x,y)dxdy

(20)

where

	$\displaystyle R(x,y)=x\left(x^{2}-a^{2}\right)[-a,0,a,x;f(x,y)]+$
	$\displaystyle+y\left(y^{2}-b^{2}\right)[-b,0,b,y;f(x,y)]-$		(21)
	$\displaystyle-x\left(x^{2}-a^{2}\right)y\left(y^{2}-b^{2}\right)\left[\begin{array}[]{l}-a,0,\quad a,x\\ -b,0,-b,y\end{array};f(x,y)\right].$

8.

Next we will seek to give an evaluation of the remainder (20).

It is observed that we can write

	$\displaystyle\rho=\int_{-a}^{a}\int_{-b}^{b}R(x,y)dxdy=\int_{-a}^{a}\left\{x(x^{2}-a^{2})\left[-a,0,a,x;\int_{-}^{b}f(x,y)dy\mid-\right.\right.$
	$\displaystyle-x\left(x^{2}-a^{2}\right)\left[-a,0,a,x\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b;f(x,y)]dy\mid\right\}dx+(22)$		(22)
	$\displaystyle+\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{\varepsilon}\right)[-b,0,b,y;f(x,y)]dxdy$

Using the elementary formula

\int_{-A}^{A}F(u)du=\int_{0}^{A}[F(t)+F(-t)]dt

(23)

we can write successively

\rho=\int_{0}^{a}\left\{t\left(t^{2}-a^{2}\right)\left[-a,0,a,t,;\int_{-b}^{b}f(t,y)dy\right]-\right.

\begin{gathered}\left.-t\left(l^{2}-a^{2}\right)\mid-a,0,a,t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\cdot[b,0,-b,y;f(t,y)]dy\right]-\\ -t\left(t^{2}-a^{2}\right)\left|a,0,-a,t;\int_{-b}^{b}f(t,y)dy\right|+\\ \left.+t\left(t^{2}-a^{2}\right)\left[a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]\right\}dt+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;/(x,y)]dtdy=\int_{0}^{a}t\left(t^{2}-a^{2}\right)([-a,0,a,t;\\ \left.\left.\int_{-b}^{b}f(t,y)dy\right]-\left[a,0,-a,t;\int_{-b}^{b}f(t,y)dy\right]\right)-t\left(t^{2}-a^{2}\right)([-a,0,a,t;\\ \left.\int_{-b}^{b}y\left(y^{2};-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]-\mid a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\\ \cdot[b,0,-b,y;/(t,y)]dy])\}+\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f(x,y)]dxdy\end{gathered}

But based on the recurrence formula of divided differences we have

\begin{gathered}\left.\mid-a,0,a,t;\int_{-b}^{b}f(t,y)dy\right]-\left[a,0,-a,-t;\int_{-b}^{b}f(t,y)dy\right]=\\ =2t\left[-a,0,a,t,-t;\int_{-b}^{b}l(t,y)dy\right]\\ {\left[-a,0,a,t;\int_{-b}^{a}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]-}\\ \left.-\mid a,0,-a,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]=\\ =2t\left[-a,0,a,t,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\right]\end{gathered}

With these we can continue writing

\begin{gathered}\rho=2\int_{0}^{a}t^{2}\left(t^{2}-a^{2}\right)\left\{\left|-a,0,a,t,-t;\int_{-b}^{b}f(t,y)dy\right|-\right.\\ -\left\{-a,0,a,t,-t;\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[b,0,-b,y;f(t,y)]dy\mid\right\}dt+\\ +\int_{-a}^{\prime\prime}\int_{-b}^{b}y\left(y^{2}-b^{-}\right)[-b,0,b,y,;f(x,y)]dxdy\end{gathered}

6 - Studies and research

Because $t^{2}\left(t^{2}-a^{2}\right)$ keeps a constant sign in the integration interval, we can apply the average formula and find

\begin{gathered}\rho=2\left\{\left[-a,0,a,\xi^{\prime},-\xi^{\prime};\int_{-b}^{b}f\left(\xi^{\prime},y\right)dy\right]-\left[-a,0,a,\xi^{\prime},-\xi^{\prime}\right.\right.\\ \left.\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;f\left(\xi^{\prime},y\right)\right]dy\right\}\int_{0}^{a}t^{2}\left(t^{2}-a^{2}\right)dt+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{2.2a^{5}}{4!15}\left(\int_{-b}^{b}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}dy-\right.\\ \left.-\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy\right)+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{a^{5}}{90}\int_{-b}^{b}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}dy+\\ +\frac{a^{5}}{90}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[b,0,-b,y;\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy+\\ +\int_{-a}^{a}\int_{-b}^{b}y\left(y^{2}-b^{2}\right)[-b,0,b,y;f]dxdy=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\\ +\int_{-b}^{b}y\left(y^{2}-b^{2}\right)\left[-b,0,b,y;\int_{-a}^{a}f(x,y)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,y)}{\partial\xi^{4}}\right]dy\end{gathered}

Applying formula (23) again, we successively obtain

		$\displaystyle\rho=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\int_{0}^{b}\left(\tau(\tau^{2}-b^{2})\left[-b,0,b,\tau;\int_{-a}^{a}f(x,\tau)dx+\right.\right.$
		$\displaystyle\left.\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]-\tau\left(\tau^{2}-b^{2}\right)\left[b,0,-b,-\tau;\int_{-a}^{a}f(x,\tau)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]\right)d\tau=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+\int_{0}^{b}\tau\left(\tau^{2}-b^{2}\right)\left(\left[-b,0,b,\tau;\int_{-a}^{a}f(x,\tau)dx+\right.\right.$
		$\displaystyle\left.\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]-\left[-b,0,b,-\tau;\int_{-a}^{a}f(x,\tau)dx+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]\right)d\tau=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+2\int_{0}^{b}\tau^{2}\left(\tau^{2}-b^{2}\right)\left[-b,0,b,\tau,-\tau;\int_{-a}^{a}f(x,\tau)dz+\right.$
		$\displaystyle\left.+\frac{a^{5}}{90}\frac{\partial^{4}f(\xi,\tau)}{\partial\xi^{4}}\right]d\tau=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}+2\left[-b,0,b,\eta^{\prime},-\eta^{\prime};\int_{-a}^{a}f\left(x,\eta^{\prime}\right)dx+\right.$

	$\displaystyle+$	$\displaystyle\left.\frac{a^{5}}{90}\frac{\partial^{4}f\left(\xi,\eta^{\prime}\right)}{\partial\xi^{4}}\right]\int_{-b}^{b}\tau^{2}\left(\tau^{2}-b^{2}\right)d\tau=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}-$
		$\displaystyle-\frac{4b^{5}}{15}\cdot\frac{1}{24}\int_{-a}^{a}\frac{\partial^{4}f(x,\eta)}{\partial\eta^{4}}dx-\frac{4a^{5}b^{5}}{15\cdot 24\cdot 90}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}=$
		$\displaystyle=-\frac{a^{5}b}{45}\frac{\partial^{4}f\left(\xi,\eta_{1}\right)}{\partial\xi^{4}}-\frac{ab^{5}}{45}\frac{\partial^{4}f\left(\xi_{1},\eta\right)}{\partial\eta^{4}}-\frac{a^{5}b^{5}}{90^{2}}\frac{\partial^{8}f(\xi,\eta)}{\partial\xi^{4}\partial\eta^{4}}$

If we introduce a symbolic swimming given by JF Steffensen [6] and more recently used by Ş. E. Mikel adze [1]

\frac{\partial^{p}f\left(\xi,\eta_{1}\right)}{\partial\xi^{p}}=D_{\xi}^{p},\frac{\partial^{q}f\left(\xi_{1},\eta\right)}{\partial\eta^{q}}=D_{\eta}^{q},\frac{\partial^{p+q}f(\xi,\eta)}{\partial\xi^{p}\eta^{q}}=D_{\xi}^{p}D_{\xi}^{q}

(24)

the remainder (20) will be written as follows

\rho=-\frac{ab}{45}\left[a^{4}D_{\xi}^{4}+b^{4}D_{\eta}^{4}+\frac{a^{4}b^{4}}{180}D_{\xi}^{4}D_{\eta}^{4}\right]

(25)

Observations. $1^{0}$ . The remainder given by Sh. E. Mike1adze on page 491 of the paper [1] must be rectified since instead of the factor $\frac{1}{180}$ which multiplies the derivative $D_{\xi}^{4}D_{\eta}^{4}$ , in its formula it appears $\frac{1}{45}$ ; this inaccuracy comes from the more general formula that precedes this one.
$2^{0}$ Formula (18), with remainder (25), was given for the square with center at the origin and side one by JF Steffensen [6], but without showing that $\xi$ and $\eta$ who intervene above are the same.

§ 2. Some practical cubature formulas for integrals $s$ -uple

9.

For the formulas we will give it will be useful to introduce an operator $S_{m}$ defined as follows

	$\displaystyle S_{i}\varphi(M)=S_{i}\varphi\left(t^{1},\ldots,\mathrm{t}^{i-1},t^{i},t^{i+1},\ldots,t^{s}\right)=$		(26)
	$\displaystyle=\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{s}\right)+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\right.$

Whatever the natural numbers are $i,k,(i<k;i,k\leqq s)$ , it is immediately established that we have

	$\displaystyle S_{i}\left(S_{k}\varphi\right)=S_{k}\left(S_{i}\varphi\right)=$
	$\displaystyle\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}+h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$
	$\displaystyle+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}+h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$		(27)
	$\displaystyle+\varphi\left(t^{1},\ldots,t^{i-1},t_{0}^{i}+h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}-h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)+$
	$\displaystyle+\rho\left(t^{1},\ldots,t^{i-1},t_{0}^{i}-h_{i}t^{i},t^{i+1},\ldots,t^{k-1},t_{0}^{k}-h_{k}t^{k},t^{k+1},\ldots,t^{s}\right)$

10.

They also need important evidence.

Lemma 1. Relative to the junction $/(M)$ and at the nodes $M_{i_{1}\ldots i_{s}}\left(t_{i_{1}}^{1},l_{i_{2}}^{2},\ldots,l_{i_{s}}^{s}\right)$ whose ordinal coordinate $\hbar$ values seem urgent

	$\displaystyle t_{0}^{k}-p_{k}h_{k},\ldots,t_{0}^{k}-h_{k},t_{0}^{k},l_{0}^{k}+h_{k},\ldots,l_{0}^{k}+p_{k}h_{k}$		(28)
	$\displaystyle(k=1,2,\ldots,s)$

we have the interpolation formula

	$\displaystyle\frac{1}{2^{s}}S_{1}S_{2}\ldots S_{s}f(M)=\frac{(-1)^{N}}{P}Uf\left(M_{0}\right)+$
	$\displaystyle+\sum_{i=1}^{s}\frac{U_{i}}{P_{i}}\sum_{j_{i}=1}^{p_{i}}(-1)^{N-j_{i}}\frac{v_{j_{i}}^{i}\left(t^{i}\right)}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!}S_{j_{i}}^{0,i_{i}}f\left(M_{0}\right)+$
	$\displaystyle+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}k=2\\ (i<k)\end{subarray}}^{s}\frac{U_{i,k}}{P_{i,k}}\sum_{j_{i}=1}^{p_{i}}\sum_{j_{k}=1}^{p_{k}}(-1)^{N-j_{i}-j_{k}}\frac{v^{i}{}_{ji}\left(t^{i}\right)}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!\left(p_{k}-j_{k}\right)!\left(p_{k}+j_{k}\right)!}$		(29)
	$\displaystyle\text{ - }S_{j_{i}}^{0,i}S_{j_{k}}^{0,k}f(Mo)$
	$\displaystyle+\sum_{j_{1}=1}^{p_{1}}\cdots\sum_{j_{s}=1}^{p_{s}}(-1)^{N-j_{1}-\cdots-j_{s}}\frac{v^{1}{}_{j_{1}}\left(l^{1}\right)}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!}\cdots\frac{v_{j_{s}}^{s}\left(l^{s}\right)}{\left(p_{s}-j_{s}\right)!\left(p_{s}+j_{s}\right)!}.$
	$\displaystyle\text{ - }S_{j_{1}}^{0,1}\ldots S_{j_{s}}^{0,s}f\left(M_{0}\right)+R_{12\ldots s}(f;M)\text{, }$

where

	$\displaystyle\left\{\begin{array}[]{l}U=u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{i}=u_{1}\left(t^{1}\right)\ldots u_{i-1}\left(t^{i-1}\right)u_{i+1}\left(t^{i+1}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{ik}=u_{1}\left(t^{1}\right)\ldots u_{i-1}\left(t^{i-1}\right)u_{i+1}\left(t^{i-1}\right)\ldots u_{k-1}\left(t^{k-1}\right)u_{k+1}\left(t^{k+1}\right)\ldots u_{s}\left(t^{s}\right)\\ U_{12}\cdots s=1;\end{array}\right.$		(30)
	$\displaystyle\left\{\begin{array}[]{l}u_{i}\left(t^{i}\right)=\left[\left(t^{i}\right)^{2}-1^{2}\right]\left[\left(t^{i}\right)^{2}-2^{2}\right]\ldots\left[\left(t^{i}\right)^{2}-p_{i}{}^{2}\right]\\ v_{ji}\left(t^{i}\right)=\left(t^{i}\right)^{2}\left[\left(t^{i}\right)^{2}-1^{2}\right]\ldots\left[\left(t^{i}\right)^{2}-\left(j_{i}-1^{2}\right)\right]\left[\left(t^{i}\right)^{2}-\left(i_{i}+1\right)^{2}\ldots\left[\left(t^{j}\right)^{2}-p_{i}{}^{2}\right];\right.\end{array}\right.$
	$\displaystyle\left\{\begin{array}[]{l}\mathrm{P}=\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{i}=\left(p_{1}!\right)^{2}\ldots\left(p_{i-1}!\right)^{2}\left(p_{i+1}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{ik}=\left(p_{1}!\right)^{2}\ldots\left(p_{i-1}!\right)^{2}\left(p_{i+1}!\right)^{2}\ldots\left(p_{k-1}!\right)^{2}\left(p_{k+1}!\right)^{2}\ldots\left(p_{s}!\right)^{2}\\ \mathrm{P}_{12}\ldots s=1;\end{array}\right.$

	$\displaystyle N=p_{1}+p_{2}+\ldots+p_{s};$		(33)
	$\displaystyle S_{j_{i}}^{0,i}f\left(M_{0}\right)=f\left(t_{0}^{1},\ldots,t_{o}^{i-1},t_{o}^{i}+j_{i}h_{i},t_{o}^{i+1},\ldots,t_{o}^{s}\right)+f\left(t_{o}^{1},\ldots,t_{o}^{i-1},t_{o}^{i}-j_{i}h_{i},t_{o}^{i+1},\ldots,t_{o}^{s}\right).$

The remainder has the following expression

R_{12}\ldots s(f;M)=\frac{1}{2s}\rho_{12}\ldots s(f;M)

with

	$\displaystyle\rho_{12}\cdots s(j;M)=2\sum_{i=1}^{s}h_{i}^{2p_{i}+2}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)\left[t_{o}^{i},l_{o}^{i}\pm h_{i},\ldots,t_{o}^{i}\pm p_{i}h_{i},l_{o}^{i}\pm h_{i}l^{i};F_{i}\right]-$
	$\displaystyle-4\sum_{\begin{subarray}{c}i=1\\ (i<k)\end{subarray}}^{s-1}\sum_{k=2}^{s}h_{i}^{2p_{i}+2}h_{k}^{2p_{k}+2}\left(t^{i}\right)^{2}\left(t^{k}\right)^{2}u_{i}\left(t^{i}\right)u_{k}\left(l^{k}\right).$		(35)

		$\displaystyle+\quad\left[\begin{array}[]{l}t_{o}^{i},t_{o}^{i}\pm h_{i},\ldots,t_{o}^{i}\pm p_{i},h_{i},t_{o}^{i}\pm h_{i}t^{i}\\ t_{o}^{k},t_{o}^{k}\pm h_{k},\ldots,t_{o}^{k}\pm p_{k}h_{k},t_{o}^{k}\pm h_{k}t^{k}\end{array}\right]F_{ik}$
		$\displaystyle+(1)^{s-1}h_{1}^{2p_{1}+2}\ldots h_{s}^{2p_{s}+2}\left(t^{1}\right)^{2}\ldots\left(t^{s}\right)^{2}u_{1}\left(t^{1}\right)\ldots u_{s}\left(t^{s}\right)\ldots$
		$\displaystyle\quad\left[\begin{array}[]{l}t_{o}^{1},t_{o}^{1}\pm h_{1},\ldots,t_{o}^{1}\pm p_{1}h_{1},t_{o}^{1}\pm h_{1}t^{1}\\ \cdots\\ t_{o}^{s},t_{o}^{s}\pm h_{s},\ldots,t_{o}^{s}\pm p_{s}h_{s},t_{o}^{s}\pm h_{n}t^{s}\end{array}\right]$
		where
		$\displaystyle\begin{array}[]{l}F_{i}=S_{1}S_{2}\ldots S_{i-1}S_{i+1}\ldots S_{s}f(M)\\ F_{ik}=S_{1}\ldots S_{i-1}S_{i+1}\ldots S_{k-1}S_{k+1}\ldots S_{s}f(M)\\ F_{12}\ldots s=f(M)\end{array}$

For the demonstration, we consider Lagrange's interpolation formula for s to be valid, with the remainder in the form given by JF Steffensen [6]. The interpolation is done on a hyperparallelepipedal network with equidistant node coordinates. That this formula can be reduced to the form (29) can be demonstrated by complete induction ¹ ).

Lemma 2. - Whatever the function $f(M)$ integrable in the hyperparallelepiped

t_{0}^{i}-mh_{i}\leqq t^{i}\leqq t_{0}^{i}+m_{i}h_{i},(i=\overline{1,s})

(37)

we have the formula

\int l_{0}^{1}+m_{1}h_{1}\\ \ldots\\ l_{0}^{1}-m_{1}h_{1}\int_{l_{0}^{s}-m_{s}h_{s}}^{l_{0}^{s}+m_{s}h_{s}}f(M)dM=h_{1}\ldots h_{s}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}S_{1}\ldots S_{s}f(M)dM.

The proof is also done by complete induction on $s$ .

⁰⁰footnotetext: ¹⁾ Vezi lucrarea [4].

11. Based on these lemmas we can state the following
theorem: Relative to the function $f(M)$ and at the nodes that formula (29) uses we have the cubature formula

$\displaystyle\iint\cdots$	$\displaystyle\int_{D}f(M)dM=\int_{t_{0}^{1}-m_{1}h_{1}}^{t_{0}^{1}+m_{1}h_{1}}\int_{t_{0}^{s}-m_{s}h_{s}}^{t_{0}^{s}+m_{s}h_{s}}f(M)dM=$
$\displaystyle=$	$\displaystyle h_{1}\ldots h_{s}\left[A_{0}^{0}f\left(M_{0}\right)+\sum_{i=1}^{s}\sum_{j_{i}=1}^{p_{i}}A_{j_{i}}^{i}S_{j_{i}}^{0,i}f\left(M_{0}\right)+\right.$	(35)
	$\displaystyle+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}k=2\\ (i<k)\end{subarray}}^{s}\sum_{j_{k}=1}^{p_{i}}\sum_{j_{k}=1}^{p_{k}}A_{j_{i}j_{k}}^{i,k}S_{j_{i}}^{0,i}S_{j_{k}}^{0,k}f\left(M_{0}\right)+$

	$\displaystyle\frac{1}{2^{s}}A_{0}^{0}=(-1)^{N}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U}{P}dM$
	$\displaystyle\frac{1}{2^{s}}A_{j_{i}}^{i}=\frac{(-1)^{N-j_{i}}}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U_{i}}{P_{i}}v_{j_{i}}^{i}dM$
	$\displaystyle\frac{1}{2^{s}}A_{j_{i}j_{k}}^{i,k}=\frac{(-1)^{N-j_{i}-j_{k}}}{\left(p_{i}-j_{i}\right)!\left(p_{i}+j_{i}\right)!\left(p_{k}-j_{k}\right)!\left(p_{k}+j_{k}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\frac{U_{ik}}{P_{ik}}v_{j_{i}}^{i}v_{j_{k}}^{k}dM$		(40)
	$\displaystyle\frac{1}{2^{s}}A_{j_{1}j_{2}\cdots j_{s}}^{1,2,\ldots,s}=\frac{(-1)^{N-j_{1}-\cdots-j_{s}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\ldots!\left(p_{s}-j_{s}\right)!\left(p_{s}+j_{s}\right)!}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}v_{j_{1}}^{1}\ldots v_{j_{s}}^{s}dM$

and

r_{s}=h_{1}\ldots h_{s}\int_{0}^{m_{1}}\ldots\int_{0}^{m_{s}}\rho_{12}\cdots_{s}(f;M)dM

(41)

Regarding the rest, we will prove the following
theorem: If in the domain $D$ function $f(M)$ it continues together $cu$ its partial derivatives of the order $\left(2p_{1}+2,2p_{2}+2,\ldots,2p_{s}+2\right)$ , then for the remainder (41) we obtain the evaluation

\frac{1}{2^{s}}r_{s}=\sum_{i=1}^{s}\frac{N_{i}h_{1}\ldots h_{i-1}h_{i}^{2p_{i}+3}h_{i+1}\ldots h_{s}}{\left(2p_{i}+2\right)!}B_{i}D_{\xi_{i}}^{2p_{i}+2}-

	$\displaystyle-\sum_{\begin{subarray}{c}i=1\\ (i<k)\end{subarray}}^{s-1}\sum_{k=1}^{s}\frac{N_{ik}h_{1}\ldots h_{i-1}h_{i}^{2p_{i}+3}h_{i+1}\ldots h_{k-1}h_{k}^{2p_{k}+3}h_{k+1}\cdots h_{s}}{\left(2p_{i}+2\right)!\left(2p_{k}+2\right)!}B_{i}B_{k}D_{\mathrm{\xi}_{i}}^{2p_{i}+2}D_{\mathrm{\xi}_{k}}^{2p_{k}+3}$
	$\displaystyle\quad+\ldots\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot h_{s}^{2p_{s}+3}$		(42)
	$\displaystyle+(-1)^{s-1}\frac{h_{1}^{2p_{1}+3}}{\left(2p_{1}+2\right)!\ldots\left(2p_{s}+2\right)!}B_{1}B_{2}\ldots B_{s}D_{\mathrm{\xi}_{1}}^{2p_{1}+2}\cdots D_{\mathrm{\xi}_{s}}^{2p_{s}+2}$

where

	$\displaystyle N_{i}=m_{1}\ldots m_{i-1}m_{i+1}\ldots m_{s}$
	$\displaystyle N_{ik}=m_{1}\ldots m_{i-1}m_{i+1}\ldots m_{k-1}m_{k+1}\ldots m_{s}$		(43)
	$\displaystyle\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$
	$\displaystyle N_{12}\ldots s=1$

and

B_{i}=\int_{0}^{m_{i}}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)dt^{i}

(44)

We will give the proof in the case $s=3$ ; in his case $s$ any one will proceed exactly as in fe1, taking only formula (35) and lemma 2 into account.

In his case $E_{3}$ the remainder (35) is written explicitly as follows

	$\displaystyle\rho_{123}(f;M)=2h_{1}^{2p_{1}+2}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]$		(45)
	$\displaystyle+2h_{2}^{2p_{2}+2}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)\left[t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}-h_{2}t^{2};F_{2}\right]$
	$\displaystyle+2h_{3}^{2p_{3}+2}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)\left[t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3};F_{0}\right]$
	$\displaystyle-4h_{1}^{2p_{1}+2}h_{2}^{2p_{2}+2}\left(t^{1}t^{2}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm h_{2}t^{2};F_{12}\end{array}\right]$
	$\displaystyle-4h_{1}^{2p_{1}+2}h_{3}^{2p_{3}+2}\left(t^{1}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{3}\left(t^{3}\right)\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{13}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$
	$\displaystyle-4h_{2}^{2p_{2}+2}h_{3}^{2p_{3}+2}\left(t^{2}t^{3}\right)^{2}u_{2}\left(t^{2}\right)u_{3}\left(t^{3}\right)\left[\begin{array}[]{l}t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}+h_{2}t^{2}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$
	$\displaystyle+8h_{1}^{2p_{1}+2}h_{2}^{2p_{2}+2}h_{3}^{2p_{3}+2}\left(t^{1}t^{2}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)u^{3}\left(t^{3}\right)\times$
	$\displaystyle\times\left[\begin{array}[]{l}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm h_{2}t^{2};F_{123}\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3}\end{array}\right]$

where

		$\displaystyle F_{1}=S_{2}S_{3}f\left(t^{1},l^{2},t^{3}\right)=f\left(t^{1},l_{0}^{2}+h_{2}t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}-h_{2}t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}+\right.$
		$\displaystyle\left.+h_{2}t^{2},t_{0}^{3}-h_{3}t^{3}\right)+f\left(t^{1},t_{0}^{2}-h_{2}t^{2},t_{0}^{3}-h_{3}t^{3}\right)$
		$\displaystyle\quad F_{2}=S_{1}S_{3}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{3}=S_{1}S_{2}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{12}=S_{3}f\left(t^{1},t^{2},t^{3}\right)=f\left(t^{1},t^{2},t_{0}^{3}+h_{3}t^{3}\right)+f\left(t^{1},t^{2},t_{0}^{3}-h_{3}t^{3}\right)$
		$\displaystyle\quad F_{13}=S_{2}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{23}=S_{1}f\left(t^{1},t^{2},t^{3}\right)$
		$\displaystyle\quad F_{123}=f\left(t^{1},t^{2},t^{3}\right).$

Applying the mean theorem of triple integrals and taking into account that $/(M)$ admits in ( $D$ ) partial derivatives of order ( $2p_{1}+2,2p_{2}+2,2p_{3}+2$ ) continue, we can write

		$\displaystyle\quad r_{3}^{1}=-h_{1}^{2p_{1}+3}h_{2}h_{3}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+h_{1}\xi_{1},t_{0}^{1}+h_{1}\xi_{1};F^{\prime}\right]_{1}+\right.$
		$\displaystyle\left.+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}\xi_{1},t_{0}^{1}-h_{1}\xi_{1};F_{1}^{\prime}\right]\right)\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{2}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)dt^{1}$

\begin{gathered}F_{1}^{\prime}=f\left(\xi_{1},t_{0}^{2}+h_{2}\eta_{11},t_{0}^{3}+h_{3}\zeta_{1}\right)+f\left(\xi_{1},t_{0}^{2}-h_{2}\eta_{1},t_{0}^{3}+h_{3}\zeta_{1}\right)+f\left(\xi_{1},t_{0}^{2}+\right.\\ \left.+h_{2}\eta_{1},t_{0}^{3}-h_{3}\zeta\right)+f\left(\xi_{1},t_{0}^{2}-h_{2}\eta_{1},t_{0}^{3}-h_{3}\zeta_{1}\right)\end{gathered}

Let's now calculate the integral

I_{3}\left(\rho_{123}\right)=h_{1}h_{2}h_{3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}\rho\left(t^{1},t^{2},t^{3}\right)dt^{1}dt^{2}dt^{3}

(46)

We will first evaluate this integral for the first term of the remainder (45); we get

		$\displaystyle r_{3}^{1}=2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]dt^{1}dt^{2}dt^{3}$
		$\displaystyle=2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{0}^{m_{1}}t^{1}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]\right)dQ_{1}\left(t^{1}\right)$

where

Q_{1}\left(t^{1}\right)=\int_{-m_{1}}^{l^{1}}t^{1}u_{1}\left(t^{1}\right)dt^{1}

is a function that according to the study by JF Steffensen [6] ² ) keeps a constant sign in the interval $\left[-m_{1},0\right]$ .

Integrating by parts we find

\begin{array}[]{r}r_{3}^{1}=-2h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)\frac{\partial}{\partial t^{1}}\left\{t^{1}\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm\right.\right.\\ \left.\pm h_{1},t^{1};F_{1}\right]dt^{1}\end{array}

	$\displaystyle{\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};F_{1}\right]}$	$\displaystyle=\frac{1}{2h_{1}t^{1}}\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+\right.\right.$
	$\displaystyle\left.+h_{1}t^{1};F_{1}\right]$	$\displaystyle\left.+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}t^{1};F_{1}\right]\right)$

so we can continue writing

	$\displaystyle r_{3}^{1}=$	$\displaystyle-h_{1}^{2p_{1}+3}h_{2}h_{3}\int_{0}^{m_{2}}dt^{2}\int_{0}^{m_{3}}dt^{3}\int_{-m_{1}}^{0}Q_{1}\left(t^{1}\right)\left(\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}+\right.\right.$
		$\displaystyle\left.\left.+h_{1}t^{1},t_{0}^{1}+h_{1}t^{1};F_{1}\right]+\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}-h_{1}t^{1}t_{0}^{1}-h_{1}t^{1};F_{1}\right]\right)dt^{1}$

⁰⁰footnotetext: ² ) pag. 155. Vezi de asemenea şi S. E. Mike1adze [1], pag, 312. ⁰⁰footnotetext: ³⁾ Vezi [4].

With these

	$\displaystyle r_{3}^{4}=-$	$\displaystyle\frac{8m_{3}h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{2}+2\right)!}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}\int_{-m_{1}}^{0}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)$
		$\displaystyle\cdot\left[t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},t_{0}^{1}\pm h_{1}t^{1};D_{\xi_{2}}^{2p_{2}+2}\right]dt^{1}$

And now, using again the evaluation given for the first term of the remainder, it is found that

r_{3}^{4}=\frac{-8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}m_{3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{1}\left(t^{1}\right)dt^{1}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}

In an analogous way, we obtain

		$\displaystyle r_{3}^{5}=\frac{-8h_{1}^{2p_{1}+3}h_{2}h_{3}^{2p_{3}+3}m_{2}}{\left(2p_{1}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{2}\left(t^{1}\right)dt^{1}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}$
		$\displaystyle r_{3}^{6}=\frac{-8h_{1}h_{2}^{2p_{3}+3}h_{3}^{2p_{3}+3}m_{1}}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{2}}^{3p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{1}\left(t^{2}\right)dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}$

A1 the seventh and last term of the remainder is

	$\displaystyle\gamma_{3}^{7}=$	$\displaystyle 8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{2}}\left(t^{1}t^{2}t^{3}\right)^{2}u_{1}\left(t^{1}\right)u_{2}\left(t^{2}\right)u_{3}\left(t^{3}\right)$
		$\displaystyle\cdot\left[\begin{array}[]{ll}t_{0}^{1},t_{0}^{1}\pm h_{1},\ldots,t_{0}^{1}\pm p_{1}h_{1},&t_{0}^{1}\pm h_{1}t^{1}\\ t_{0}^{2},&t_{0}^{2}\pm h_{2},\ldots,\\ t_{0}^{2}\pm p_{2}h_{2},&t_{0}^{2}\pm h_{2}t^{2};/\\ t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},&t_{0}^{3}\pm h_{3}t^{3}\end{array}\right].dt^{1}dt^{2}dt^{3}$

Based on the previous we have

\begin{gathered}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)\left[t_{0}^{3},t_{0}^{3}\pm h_{3},\ldots,t_{0}^{3}\pm p_{3}h_{3},t_{0}^{3}\pm h_{3}t^{3};f\right]dt^{3}=\\ =\frac{1}{\left(2p_{3}+2\right)!}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\end{gathered}

and

\begin{gathered}\frac{1}{\left(2p_{3}+2\right)!}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\int_{0}^{m_{3}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)\left[t_{0}^{2},t_{0}^{2}\pm h_{2},\ldots,t_{0}^{2}\pm p_{2}h_{2},t_{0}^{2}\pm\right.\\ \left.\quad\pm h_{2}t^{2};D_{\xi_{3}}^{2p_{3}+2}\right]dt^{2}=\frac{1}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\times\\ \times\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}\left(t^{2}\right)dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}\left(t^{3}\right)dt^{3}\end{gathered}

So that

\begin{gathered}r_{3}^{7}=\frac{8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}\int_{0}^{m_{1}}\left(t^{1}\right)^{2}u_{1}dt^{1}\times\\ \times\int_{0}^{m_{2}}\left(t^{2}\right)^{2}u_{2}dt^{2}\int_{0}^{m_{3}}\left(t^{3}\right)^{2}u_{3}dt^{3}\end{gathered}

So in $E_{3}$ the rest of the cubature formula (39) can be expressed by the formula

$\displaystyle\gamma_{3}$	$\displaystyle=\gamma_{3}^{1}+\ldots+\gamma_{3}^{7}=\frac{8m_{2}m_{3}h_{1}^{2p_{1}+3}h_{2}h_{3}}{\left(2p_{1}+2\right)!}B_{1}D_{\xi_{1}}^{2p_{1}+2}+$
$\displaystyle+$	$\displaystyle\frac{8m_{1}m_{3}h_{1}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{2}+2\right)!}B_{2}D_{\xi_{2}}^{2p_{2}+2}+\frac{8m_{1}m_{2}h_{1}h_{2}h_{3}^{2p_{3}+3}}{\left(2p_{3}+2\right)!}B_{3}D_{\xi_{3}}^{2p_{3}+2}-$
	$\displaystyle-\frac{8m_{3}h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!}B_{1}B_{2}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}-$
	$\displaystyle-\frac{8m_{2}h_{1}^{2p_{1}+3}h_{2}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{3}+2\right)!}B_{1}B_{3}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{3}}^{2p_{3}+2}-$	(47)
	$\displaystyle-\frac{8m_{1}h_{1}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}B_{2}B_{3}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}+$
$\displaystyle+$	$\displaystyle\frac{8h_{1}^{2p_{1}+3}h_{2}^{2p_{2}+3}h_{3}^{2p_{3}+3}}{\left(2p_{1}+2\right)!\left(2p_{2}+2\right)!\left(2p_{3}+2\right)!}B_{1}B_{2}B_{3}D_{\xi_{1}}^{2p_{1}+2}D_{\xi_{2}}^{2p_{2}+2}D_{\xi_{3}}^{2p_{3}+2}$

where

B_{i}=\int_{0}^{m_{i}}\left(t^{i}\right)^{2}u_{i}\left(t^{i}\right)dt^{i}

12.

An important category of cubature formulas is obtained from (39) if we take $m_{i}=p_{i},i=\overline{1,s}$ We will call such formulas, together with IF Steffensen [6] and SE Mikeladze [1], closed-type formulas.

In the case where the limits of the integral of $i-a(i=\overline{1,s})$ are out of range ( $t_{0}^{i}-p_{i}h_{i},t_{0}^{i}+p_{i}h_{i}$ ), in other words $m_{i}>p_{i}(i=\overline{1,s})$ , we obtain the so-called open-type formulas.

And finally, if $m_{i}<p_{i}$ , we obtain the cubature formulas with nodes placed outside the integration domain.

§ 3. Important particular cases of the previous cubature formulas

13.

We will now consider certain particular cases, which seem more interesting to us, of formula (39).

In case $s=1$ most of the formulas that are obtained were given by IF Steffensen [6], SE Mikeladze [1], WE Milne [2], etc.

Changing the notations slightly, formula (39) in the case $s=1$ BECOMES

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=h\left[A_{0}^{0}f\left(x_{0}\right)+\sum_{j=1}^{p}A_{j}^{1}S_{j}^{0}f\left(x_{0}\right)\right]+r_{1}=$		(48)
	$\displaystyle=h\left[A_{0}^{0}f\left(x_{0}\right)+\sum_{j=1}^{p}A_{j}^{1}\left(f\left(x_{0}+jh\right)+f\left(x_{0}-jh\right)\right)\right]+r_{1}$

where

	$\displaystyle\frac{1}{2}A_{0}^{0}=\frac{(-1)^{p}}{(p!)^{2}}\int_{0}^{m}\left(x^{2}-1\right)\ldots\left(x^{2}-p^{2}\right)dx$
	$\displaystyle\frac{1}{2}A_{j}^{0}=\frac{(-1)^{p-j}}{(p-j)!(p+j)!}\int_{0}^{m}x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-i-1^{2}\right)$		(52)

and

\frac{1}{2}r_{1}=\frac{h^{2p+3}}{(2p+2)!}D_{\xi}^{2p+2}\int_{0}^{m}x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-p^{2}\right)dx

14.

If we take (48) $p=0$ the quadrature formula with one node is obtained

\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=2mhf\left(x_{0}\right)+\frac{m^{3}h^{3}}{3}f^{\prime\prime}(\xi)

(49)

and if we do $p=1$ the 3-node formula is found

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}f(x)dx=\frac{mh}{3}\left[2\left(3-m^{2}\right)f\left(x_{0}\right)+m^{2}f\left(x_{0}+h\right)+f\left(x_{0}-h\right)\right]$		(50)
	$\displaystyle+\frac{m^{3}\left(3m^{2}-5\right)}{180}h^{5}f(IV)(\xi)$

For $m=1$ a well-known closed-type formula is obtained: the Cavalieri-Simpson formula.

For $m=3,p=2$ the open-type quadrature formula is found

	$\displaystyle\int_{x_{0}-3h}^{x_{0}+3h}f(x)dx=\frac{h}{30}\left[234f\left(x_{0}\right)-126\overline{f\left(x_{0}+h\right)+f\left(x_{0}-h\right)}+\right.$		(54)
	$\displaystyle\left.\quad+99\overline{f\left(x_{0}+2h\right)+f\left(x_{0}-2h\right)}\right]+\frac{41}{140}h^{7}f(VI)(\xi)$

of accuracy level 5.
15. In the case of $\therefore=2$ the cubature formula (39) is written

	$\displaystyle\int_{x_{0}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=hk\left[A_{0}^{0}f\left(x_{0},x_{0}\right)+\sum_{i=1}^{p}A_{i}^{1}S_{i}^{0,1}f\left(x_{0},y_{0}\right)+\right.$
	$\displaystyle\left.+\sum_{j=1}^{q}A_{j}^{2}S_{j}^{0,2}f\left(x_{0},y_{0}\right)+\sum_{i=1}^{p}\sum_{j=1}^{q}A_{i,j}^{1,2}S_{i}^{0,1}S_{j}^{0,2}f\left(x_{0},y_{0}\right)\right]+r_{2},$		(51)

or more explicitly

\int_{x_{u}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=

		$\displaystyle=hk\left[A_{0}^{0}f\left(x_{0},y_{0}^{\prime}\right)+\sum_{i=1}^{p}A_{i}^{1}\left(f\left(x_{0}+ih,y_{0}\right)+f\left(x_{0}-ih,y_{0}\right)\right)+\right.$
		$\displaystyle\quad+\sum_{j=1}^{q}A_{j}^{2}\left(f\left(x_{0},y_{0}+jk\right)+f\left(x_{0},y_{0}-jk\right)\right)+$
		$\displaystyle+\sum_{i=1}^{p}\sum_{j=1}^{q}A_{i,j}^{1,2}\left(f\left(x_{0}+ih,y_{0}+jk\right)+f\left(x_{0}-ih,y_{0}+jk\right)+\right.$
		$\displaystyle\left.\left.\quad+f\left(x_{0}+ih,y_{0}-jk\right)+i\left(x_{0}-ih,y_{0}-jk\right)\right)\right]+r_{2},$

where

\begin{array}[]{r}\frac{1}{4}A_{0}^{0}=\frac{(-1)^{p+q}}{(p!)^{2}(q!)^{2}}\int_{0}^{m}\int_{0}^{n}u(x)v(y)dxdy^{\prime}\\ \frac{1}{4}A_{i}^{1}=\frac{(-1)^{p+q-i}}{(p-i)!(p+i)!(q!)^{2}}\int_{0}^{m}\int_{0}^{n}v(y)a_{i}(x)dxdy\\ \frac{1}{4}A_{j}^{2}=\frac{(-1)^{p+q-j}}{(p!)^{2}(q-j)!(q+j)!}\int_{0}^{m}\int_{0}^{n}u(x)b_{j}(y)dxdy\\ \frac{1}{1}A_{i,j}^{1,2}=\frac{(-1)^{p+q-i-j}}{(p-i)!(p+i)!(q-j)!(q+j)!}\int_{0}^{m}\int_{0}^{n}a_{i}(x)b_{j}(y)dxdy\end{array}

and

\begin{gathered}\|(x)=\prod_{i=1}^{p}\left(x^{2}-i^{2}\right),\quad v(y)=\prod_{j=1}^{q}\left(y^{2}-j^{2}\right)\\ l_{i}(x)=x^{2}\left(x^{2}-1\right)\ldots\left(x^{2}-\bar{i}-1^{2}\right)\left(x^{2}-\overline{i+1^{2}}\right)\ldots\left(x^{2}-p^{2}\right)\\ b_{j}(y)=y^{2}\left(y^{2}-1\right)\ldots\left(y^{2}-\overline{j-1^{2}}\right)\left(y^{2}-\overline{j+1^{2}}\right)\ldots\left(y^{2}-q^{2}\right)\end{gathered}

The remainder has the following expression

$\displaystyle\frac{1}{4}r_{2}$	$\displaystyle=\frac{nh^{:n+3}k}{(2p+2)!}D_{\mathrm{E}}^{2p+2}\int_{0}^{m}x^{2}u(x)dx+{}_{(2q+2)!}^{mhk^{2q+3}}D_{\eta}^{2q+2}\int_{0}^{n}y^{2}v(y)dy-$
	$\displaystyle\quad h^{2p+3}k^{2q+3}$	(55)
	$\displaystyle\left(2p+2!(2q+2)!D_{\mathrm{E}}^{2p+2}D_{\eta}^{2q+2}\int_{0}^{m}x^{2}u(x)dx\int_{0}^{n}y^{2}v(y)dy\right.$

16.

If in (52) we take $p=q=0$ the following open-type cubature formula of accuracy degree is obtained $(1,1)$

\int_{x_{n}-mh}^{x_{0}+mh}dx\int_{y_{n}-nk}^{y_{n}+nk}f\left(x,ydy^{\prime}=4mnhkf\left(x_{0},y_{0}\right)=\rho\right.

where

\rho=\frac{2}{3}m^{3}nh^{3}kD_{\xi}^{2}+\frac{2}{3}mn^{3}hk^{3}D_{\eta}^{2}-\frac{1}{9}m^{3}n^{3}h^{3}k^{3}D_{\xi}^{2}D_{\xi}^{2}

An important cubature formula is obtained if we take $p=q=1$

\begin{gathered}\int_{x_{0}-mh}^{x_{0}+mh}dx\int_{y_{0}-nk}^{y_{0}+nk}f(x,y)dy=\\ =\frac{nmhk}{9}\left[4\left(m^{2}-3\right)\left(n^{2}-3\right)f\left(x_{0},y_{0}\right)-2m^{2}\left(n^{3}-3\right)\right.\\ \cdot\left(f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)\right)-2n^{2}\left(m^{2}-3\right)\left(f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right)+\\ +m^{2}n^{2}\left(f\left(x_{0}+h,y_{0}+k\right)\right.\\ \left.+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right]+\rho\end{gathered}

where the remainder has the expression

\begin{gathered}\rho=\frac{h^{5}km^{3}n\left(3m^{2}-5\right)}{90}D_{\xi}^{4}+\frac{hk^{5}mn^{3}\left(3n^{2}-5\right)}{90}D_{\eta}^{4}-\\ -\frac{h^{5}k^{5}m^{3}n^{3}\left(3m^{2}-5\right)\left(3n^{2}-5\right)}{144}D_{\xi}^{4}D_{\eta}^{4}.\end{gathered}

Doing above $m=n=1$ we arrive at the Cavalieri-Simpson cubature formula for two variables

\begin{gathered}\qquad\int_{x_{o}-h}^{x_{o}+h}dx\int_{y_{o}-k}^{y_{o}+k}f(x,y)dy=\\ =\frac{hk}{9}\left[f\left(x_{0}+h,y_{0}+k\right)+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right.\\ \left.+4\left(f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)+f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right)+16f\left(x_{0},y_{0}\right)\right]+\rho\\ \text{ unde }\end{gathered}

\rho=-\frac{hk}{45}\left[h^{4}D_{\xi}^{4}+k^{4}D_{\eta}^{4}+\frac{1}{180}h^{4}k^{4}D_{\xi}^{4}D_{\eta}^{4}\right].

We thus found the expression (25) of the remainder of the Cavalieri-Simpson formula in another way.

For $n=n=2$ HAVE

\begin{gathered}\int_{x_{0}-2h}^{x_{0}+2h}dx\int_{y_{0}-2k}^{y_{0}+2k}f(x,y)dy=\\ =\frac{16hk}{9}\left\{f\left(x_{0},y_{0}\right)-2\left[f\left(x_{0}+h,y_{0}\right)+f\left(x_{0}-h,y_{0}\right)+f\left(x_{0},y_{0}+k\right)+f\left(x_{0},y_{0}-k\right)\right]+\right.\\ \left.+4\left[f\left(x_{0}+h,y_{0}+k\right)+f\left(x_{0}-h,y_{0}+k\right)+f\left(x_{0}+h,y_{0}-k\right)+f\left(x_{0}-h,y_{0}-k\right)\right]\right\}+\rho\end{gathered}

with

\rho=\frac{56}{45}h^{5}kD_{\xi}^{4}+\frac{56}{45}hk^{5}D_{\eta}^{4}-\frac{196}{9}h^{5}k^{5}D_{\xi}^{4}D_{\eta}^{4}.

For $p=q=2$ formula (52) becomes, taking for simplification $x_{0}=y_{0}=0$

	$\displaystyle\quad\int_{-mh}^{mh}\int_{-nk}^{nk}f(x,y)dxdy=$
	$\displaystyle=\frac{nmhk}{32400}\left\{C_{00}f(0,0)+C_{10}[f(h,0)+f(-h,0)]+C_{20}[f(2h,0)+f(-2h,0)]\right.$		(56)
	$\displaystyle+C_{01}[f(0,k)+f(0,-k)]+C_{62}[f(0,2k)+f(0,-2k)]+C_{11}[f(h,h)+f(-h,k)$
	$\displaystyle+f(h,-k)+f(-h,-k)]+C_{12}[f(h,2k)+f(-h,2k)+f(h,-2k)+f(-h,-2k)]$
	$\displaystyle+C_{21}[f(2h,k)+f(-2h,k)+f(2h,-k)+f(-2h,-k)]+C_{22}[f(2h,2k)$
	$\displaystyle+f(-2h,2k)+f(2h,-2k)+f(-2h,-2k)]\}+\rho.$

where

	$\displaystyle\rho$	$\displaystyle=\frac{nmhk}{3780}\left[Ah^{6}D_{\xi}^{6}+Bk^{6}D_{\eta}^{6}-Ch^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right]$
	$\displaystyle C_{00}$	$\displaystyle=6\left(3m^{4}-25m^{2}+60\right)\left(3n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{10}$	$\displaystyle=-4m^{2}\left(3m^{2}-20\right)\left(2n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{20}$	$\displaystyle=6m^{2}\left(3m^{2}-5\right)\left(3n^{4}-25n^{2}+60\right)$
	$\displaystyle C_{01}$	$\displaystyle=-4n^{2}\left(3n^{2}-20\right)\left(3m^{4}-25m^{2}+60\right)$
	$\displaystyle C_{02}$	$\displaystyle=6n^{2}\left(3n^{2}-5\right)\left(3m^{4}-25n^{2}+60\right)$
	$\displaystyle C_{11}$	$\displaystyle=6m^{2}n^{2}\left(3m^{2}-20\right)\left(3n^{2}-20\right)$
	$\displaystyle C_{12}$	$\displaystyle=-4m^{2}n^{2}\left(3m^{2}-20\right)\left(3n^{2}-5\right)$
	$\displaystyle C_{21}$	$\displaystyle=-4m^{2}n^{2}\left(3m^{2}-5\right)\left(3n^{2}-20\right)$
	$\displaystyle C_{22}$	$\displaystyle=m^{2}n^{2}\left(3m^{2}-5\right)\left(3n^{2}-5\right)$

and

		$\displaystyle A=m^{2}\left(3m^{4}-21m^{2}+28\right)$
		$\displaystyle B=n^{2}\left(3n^{4}-21n^{2}+28\right)$
		$\displaystyle C=\frac{1}{15120}m^{2}n^{2}\left(3m^{4}-21m^{2}+28\right)\left(3n^{4}-21n^{2}+28\right)$

If in (56) it is done $n=m=2$ the following closed-type cubature formula is obtained, which has a partial degree of accuracy $(5,5)$ :
$\int_{-2h}^{2h}\int_{-2k}^{2k}f(x,y)dxdy=\frac{4hk}{2025}\{144f(0,0)+384[f(-h,0)+f(h,0)+f(0,-k)+f(0,k)]+84[f(-2h,0)+f(0,-2k)+f(0,2k)+f(2h,0)]+1024[f(-h,-k)+f(-h,k)++f(h,-k)+f(h,k)]+224[f(-2h,-k)+f(-2h,k)+f(-h,-2k)+f(-h,2k)++f(h,-2k)+f(h,2k)+f(2h,-k)+f(2h,k)]+49[f(-2h,-2k)+f(-2h,2k)+f(2h,-2k)+f(2h,2k)]\}+\mathrm{p}$ with

\rho=-\frac{32hk}{945}\left[h^{6}D_{\xi}^{6}+k^{6}D_{\eta}^{6}+\frac{2}{945}h^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right].

For $m=n=3$ the open type cubature formula is obtained

		$\displaystyle\quad\int_{-3h}^{3h}\int_{-3k}^{3k}f(x,y)dxdy=\frac{9hk}{100}\{76f(0,0)-64[f(h,0)+f(-h,0)+f(0,k)$
		$\displaystyle+f(0,-k)]+86[f(0,2k)+f(0,-2k)+f(2h,0)]+96[f(h,k)+$
		$\displaystyle+f(-h,k)+f(h,-k)+f(-h,-k)]-54[f(h,2k)+f(-h,2k)+f(h,-2k)$
		$\displaystyle+f(-h,-2k)+f(2h,k)+f(-2h,k)+f(2h,-k)+f(-2h,-k)]+21\quad[f(2h,2k)$
		$\displaystyle+f(-2h,2k)+f(2h,-2k)+f(-2h,-2k)]+p$

where

\rho=\frac{123}{70}hk\left[h^{6}D_{\xi}^{6}+k^{6}D_{\eta}^{6}-\frac{41}{840}h^{6}k^{6}D_{\xi}^{6}D_{\eta}^{6}\right]

For $m=n=2$ from (56) we obtain a cubature formula with nodes outside the integration domain, which is worth mentioning.
17. In the case $s=3$ the cubature formula (39) is written

\begin{gathered}\int_{x_{0}-m_{1}h_{1}}^{x_{0}+m_{1}h_{1}}dx\int_{y_{0}-m_{2}h_{2}}^{y_{0}+m_{2}h_{2}}dy\int_{z_{0}-m_{3}h_{3}}^{z_{0}+m_{3}h_{3}}f(x,y,z)dz=\\ =h_{1}h_{2}h_{3}\left[A_{0}^{0}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{1}=1}^{p_{1}}A_{j_{1}}^{1}S_{j_{1}}^{0,1}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{2}=1}^{p_{2}}A_{j_{2}}^{2}S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)\right.\\ +\sum_{j_{3}=1}^{p_{3}}A_{j_{3}}^{3}S_{j_{3}}^{0,3}/\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{1}=1}^{p_{1}}\sum_{j_{2}=1}^{p_{0}}A_{j_{1}j_{2}}^{1,2}S_{j_{1}}^{0,1}S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)+\\ +\sum_{j_{1}=1}^{p_{1}}\sum_{j_{3}=1}^{p_{3}}A_{j_{1}j_{3}}^{1,3}S_{j_{1}}^{0,1}S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)+\sum_{j_{2}=1}^{p_{2}}\sum_{j_{3}=1}^{p_{3}}A_{j_{2}j_{3}}^{2,3}S_{j_{2}}^{0,2}S_{j_{3}}^{0,3}f\left(x_{0},y_{0}z_{0}\right)\\ \left.+\sum_{j_{1}=1}^{p_{1}}\sum_{j_{2}=1}^{p_{3}}\sum_{j_{3}=1}^{p_{4}}A_{j_{1}j_{2}j_{3}}^{1,2,3}S_{j_{1}}^{0,1}S_{j_{2}}^{0,2}S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)\right]+\gamma_{3},\end{gathered}

MIND

$\displaystyle S_{j_{1}}^{0,1}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0}+j_{1}h_{1},y_{0},z_{0}\right)+f\left(x_{0}-j_{1}h_{1},y_{0},z_{0}\right)$	(58)
$\displaystyle S_{j_{2}}^{0,2}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0},y_{0}+j_{2}h_{2},z_{0}\right)+f\left(x_{0},y_{0}-j_{2}h_{2},z_{0}\right)$
$\displaystyle S_{j_{3}}^{0,3}f\left(x_{0},y_{0},z_{0}\right)$	$\displaystyle=f\left(x_{0},y_{0},z_{0}+j_{3}h_{3}\right)+/\left(x_{0},y_{0},z_{0}-j_{3}h_{3}\right)$

and the rest $r_{3}$ has the expression from (42) with the modification of the notation already used

\begin{array}[]{lll}t^{1}=x,&t^{2}=y,&t^{3}=z\\ t_{0}^{1}=x_{0},&t_{0}^{2}=y_{6},&t_{0}^{3}=z_{0}\end{array}

The coefficients of formula (57) have the expressions
$\frac{1}{8}A_{0}^{0}=\frac{(-1)^{p_{1}+p_{2}+p_{3}}}{\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}\left(p_{3}!\right)}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)u_{2}(y)u_{3}(z)dxdydz$
$\frac{1}{8}A_{j_{1}}^{1}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}!\right)^{2}\left(p_{3}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)u_{2}(y)u_{3}(z)dxdydz$
$\frac{1}{8}A_{j_{2}}^{2}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{2}}{\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{1}!\right)^{2}\left(p_{3}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)v_{j_{3}}^{2}(y)u_{3}(z)dxdzdy$
$\frac{1}{8}A_{j_{3}}^{3}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{3}}}{\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!\left(p_{1}!\right)^{2}\left(p_{2}!\right)^{2}}\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)u_{2}(y)v_{j_{3}}^{3}(z)dxdydz$
$\frac{1}{8}A_{j_{1}j_{2}}^{1,2}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}-j_{2}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}!\right)^{2}}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)v_{j_{2}}^{2}(y)u_{3}(z)dxdydz

$\frac{1}{8}A_{j_{1}j_{3}}^{1,3}=\frac{(-1)^{p_{1}+p_{2}+p_{3}-j_{1}-j_{3}}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}!\right)^{2}\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m^{3}}v_{j_{1}}^{1}(x)u_{2}(y)v_{j_{3}}^{3}(z)dxdydz

$\frac{1}{8}A_{j_{2}j_{3}}^{2,3}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{2}-j_{3}}{\left(p_{1}!\right)^{2}\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}-j_{3}\right)!\left(p_{3}+i_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}u_{1}(x)v_{j_{2}}^{2}(y)v_{j_{3}}^{3}(z)dxdydz

$\frac{1}{8}A_{j_{1}j_{2}j_{2}}^{1,2,3}=\frac{(-1)^{p_{1}}+p_{2}+p_{3}-j_{1}-j_{2}-j_{3}}{\left(p_{1}-j_{1}\right)!\left(p_{1}+j_{1}\right)!\left(p_{2}-j_{2}\right)!\left(p_{2}+j_{2}\right)!\left(p_{3}-j_{3}\right)!\left(p_{3}+j_{3}\right)!}$ .

\int_{0}^{m_{1}}\int_{0}^{m_{2}}\int_{0}^{m_{3}}v_{j_{1}}^{1}(x)v_{j_{2}}^{2}(y)v_{j_{3}}^{3}(z)dxdydz

18.

We will now focus on some important particular cases of this formula.

For $p_{1}=p_{2}=p_{3}=0$ a cubature formula is obtained that uses a single node and has a partial degree of accuracy ( $1,1,1$ )

\iiint_{D}f(x,y,z)dxdydz=8m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}f\left(x_{0},y_{0},z_{0}\right)+\rho

(59)

where $D$ is the parallelepiped

x_{0}-m_{1}h_{1}\leqq x\leqq x_{0}+m_{1}h_{1},y_{0}-m_{2}h_{2}\leqq y\leqq y_{0}+m_{2}h_{2},z_{0}-m_{3}h_{3}\leqq z\leqq z_{0}+m_{3}h_{3}

(60)

and the rest has the expression.
$\rho=\frac{m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}}{3}\left\lceil 4m_{1}^{2}h_{1}^{2}D_{\xi}^{2}+4m_{2}^{2}h_{2}^{2}D_{\eta}^{2}+4m_{3}^{2}h_{3}^{3}D_{\xi}^{2}-\frac{2}{3}m_{1}^{2}m_{2}^{2}h_{1}^{2}h_{2}^{2}D_{\xi}^{2}D_{\eta}^{2}-\right.$
$\left.-\frac{2}{3}m_{1}^{2}m_{3}^{2}h_{1}^{2}h_{3}^{2}D_{\xi}^{2}D_{\zeta}^{2}-\frac{2}{3}m_{2}^{2}m_{3}^{2}h_{2}^{2}h_{3}^{2}D_{\eta}^{2}D_{\xi}^{2}+\frac{1}{9}m_{1}^{2}m_{2}^{2}m_{3}^{2}h_{1}^{2}h_{2}^{2}h_{3}^{2}D_{\xi}^{2}D_{\eta}^{2}D_{\xi}^{2}\right]$
It is observed that the only node on which this formula is defined is found in the center of gravity of the domain . $D$ , assumed homogeneous. This formula is Gaussian, since it uses the minimum possible number of nodes.
19. Substituting into formula (57) $p=p_{2}=p_{3}=1$ , the cubature formula is obtained

	$\displaystyle\iiint_{D}f(x,y,z)dxdydz=$
	$\displaystyle=\frac{m_{1}m_{2}m_{3}h_{1}h_{2}h_{3}}{27}\int-8\left(m_{1}^{2}-3\right)\left(m_{2}^{2}-3\right)\left(m_{3}^{2}-3\right)f\left(x_{0},y_{0},z_{0}\right)+4m_{1}^{2}\left(m_{2}^{2}-3\right)\left(m_{3}^{2}-3\right)$
	$\displaystyle.S_{1}^{0,1}f\left(x_{0}y_{0},z_{0}\right)+4\left(m_{1}^{2}-3\right)m_{2}^{2}\left(m_{3}^{2}-3\right)S_{1}^{0,2}f\left(x_{0},y_{0},z_{0}\right)+4\left(m_{1}^{2}-3\right)\left(m_{2}^{2}-3\right)m_{3}^{2}$
	$\displaystyle.S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)-2m_{1}^{2}m_{2}^{2}\left(m_{3}^{2}-3\right)S_{1}^{0,1}S_{1}^{0,2}f\left(x_{0},y_{0},z_{0}\right)-2m_{1}^{2}\left(m_{2}^{2}-3\right)m_{3}^{2}$		(60)
	$\displaystyle.S_{1}^{0,1}S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)-2\left(m_{1}^{2}-3\right)m_{2}^{2}m_{3}^{2}S_{1}^{0,2}S_{1}^{0,3}f\left(x_{0},y_{0},z_{0}\right)+m_{1}^{2}m_{2}^{2}m_{3}^{2}$

\begin{gathered}\rho=\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}m_{3}h_{1}^{5}h_{2}h_{3}}{45}D_{5}^{4}+\frac{m_{1}m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}h_{1}h_{2}^{5}h_{3}}{45}D_{\eta}^{4}\\ +\frac{m_{1}m_{2}m_{3}^{3}\left(3m_{3}^{2}-5\right)h_{1}h_{2}h_{3}^{5}}{45}D_{5}^{4}-\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}}{16200}h_{1}^{5}h_{2}^{5}h_{3}D_{5}^{4}D_{\eta}^{4}\\ -\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}m_{3}^{3}\left(3m_{3}^{2}-5\right)}{16200}h_{1}^{5}h_{2}h_{3}^{5}D_{\xi}^{4}D_{5}^{4}-\frac{m_{1}m_{2}^{3}\left(3m_{2}^{2}-5\right)m_{3}^{3}\left(3m_{3}^{2}-5\right)}{16200}\\ \cdot h_{1}h_{2}^{5}h_{3}^{5}D_{\eta}^{4}D_{5}^{4}+\frac{m_{1}^{3}\left(3m_{1}^{2}-5\right)m_{2}^{2}\left(3m_{2}^{2}-5\right)m_{3}^{3}\left(3m_{3}^{2}-5\right)}{5832000}h_{1}^{5}h_{2}^{5}h_{3}^{5}D_{5}^{4}D_{\eta}^{4}D_{5}^{4}.\end{gathered}

From this we immediately obtain the following cubature formula which represents the extension of the Cavalieri-Simpson formula to three variables

		$\displaystyle\quad\int_{x_{0}-h_{1}}^{x_{0}+h_{1}}dx\int_{y_{0}-h_{2}}^{y_{0}+h_{2}}dy\int_{z_{0}-h_{3}}^{z_{0}+h_{3}}f(x,y,z)dz=$
		$\displaystyle=\frac{h_{1}h_{2}h_{3}}{27}\left\{f\left(x_{0}+h_{1},y_{0}+h_{2},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0}-h_{1},y_{0}+h_{2}z_{0}+h_{3}\right)\right.$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}-h_{3}\right)$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}-h_{3}\right)+4\left[f\left(x_{0},y_{0}+h_{2},z_{0}+h_{3}\right)+\right.$
		$\displaystyle+f\left(x_{0},y_{0}+h_{2},z_{0}-h_{3}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}+h_{3}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}-h_{3}\right)+$
		$\displaystyle+f\left(x_{0}+h_{1},y_{0},z_{0}+h_{3}\right)+f\left(x_{0}+h_{1},y_{0},z_{0}-h_{3}\right)+f\left(x_{0}-h_{1},y_{0},z_{0}+h_{3}\right)+$
		$\displaystyle+f\left(x_{0}-h_{1},y_{0},z_{0}-h_{3}\right)+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}-h_{1},y_{0}+h_{2},z_{0}\right)$
		$\displaystyle\left.+f\left(x_{0}+h_{1},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}-h_{1},y_{0}-h_{2},z_{0}\right)\right]+6\left[f\left(x_{0},y_{0},z_{0}+h_{3}\right)+\right.$
		$\displaystyle+f\left(x_{0},y_{0},t_{0}-h_{3}\right)\Omega f\left(x_{0},y_{0}+h_{2},z_{0}\right)+f\left(x_{0},y_{0}-h_{2},z_{0}\right)+f\left(x_{0}+h_{1},y_{0},z_{0}\right)+$
		$\displaystyle\left.\left.+f\left(x_{0}-h_{1},y_{0},z_{0}\right)\right]+64f\left(x_{0},y_{0},z_{0}\right)\right\}+\rho$

where

\begin{gathered}\rho=-\frac{h_{1}h_{2}h_{3}}{45}\left[2h_{1}^{4}D_{\xi}^{4}+2h_{2}^{4}D_{\eta}^{4}+2h_{3}^{4}D_{\xi}^{4}+\frac{1}{90}h_{1}^{4}h_{2}^{4}D_{\xi}^{4}D_{\eta}^{4}+\frac{1}{90}h_{1}^{4}h_{3}^{4}D_{\xi}^{4}D_{\xi}^{4}+\right.\\ \left.+\frac{1}{90}h_{2}^{4}h_{3}^{4}D_{\eta}^{4}D_{\xi}^{4}+\frac{1}{16200}h_{1}^{4}h_{2}^{4}h_{3}^{4}D_{\xi}^{4}D_{\eta}^{4}D_{\xi}^{4}\right]\end{gathered}

If in formula (60) we do $m_{1}=m_{2}=m_{3}=2$ the following open-type cubature formula is obtained, which uses the same number of nodes as formula (61) and has the same partial degree of accuracy ( $3,3,3$ )

	$\displaystyle\quad\int_{-2h_{1}}^{2h_{1}}\int_{-2h_{2}}^{2h_{2}}\int_{-2h_{3}}^{2h_{3}}f(x,y,z)dxdydz=$
	$\displaystyle=\frac{64h_{1}h_{2}h_{3}}{27}\left\{-f(0,0,0)+2\left[f\left(0,0,h_{3}\right)+f\left(0,0,-h_{3}\right)+f\left(0,h_{2},0\right)+f\left(0,-h_{2},0\right)+\right.\right.$
	$\displaystyle\left.+f\left(h_{1},0,0\right)+f\left(-h_{1},0,0\right)\right]-4\left[f\left(0,h_{2},h_{3}\right)+f\left(0,h_{2},-h_{3}\right)+f\left(0,-h_{2},h_{3}\right)\right.$		(62)
	$\displaystyle+f\left(0,-h_{2},-h_{3}\right)+f\left(h_{1},0,h_{3}\right)+f\left(h_{1},0,-h_{3}\right)+f\left(-h_{1},0,h_{3}\right)+f\left(-h_{1},0,-h_{3}\right)+$
	$\displaystyle\left.+f\left(h_{1},h_{2},0\right)+f\left(-h_{1},h_{2},0\right)+f\left(h_{1},-h_{2},0\right)+f\left(-h_{1},-h_{2},0\right)\right]+8\left[f\left(h_{1},h_{2},h_{3}\right)+\right.$
	$\displaystyle+f\left(h_{1},h_{2},-h_{3}\right)+f\left(-h_{1},h_{2},h_{3}\right)+f\left(-h_{1},h_{2},-h_{3}\right)+f\left(h_{1},-h_{2},h_{3}\right)+f\left(h_{1},-h_{2},-h_{3}\right)+$
	$\displaystyle\left.\left.+f\left(-h_{1},-h_{2},h_{3}\right)+f\left(-h_{1},-h_{2},-h_{3}\right)\right]\right\}+\rho,$

\begin{gathered}\rho=\frac{1792}{45}h_{1}h_{2}h_{3}\left[4\left(h_{1}^{4}D_{\xi}^{4}+h_{2}^{4}D_{\eta}^{4}+h_{3}^{4}D_{\xi}^{4}\right)-\frac{98}{45}\left(h_{1}^{4}h_{2}^{4}D_{\xi}^{4}D_{\eta}^{4}+h_{1}^{4}h_{3}^{4}D_{\xi}^{4}D_{\xi}^{4}+\right.\right.\\ \left.\left.+h_{2}^{4}h_{3}^{4}D_{\eta}^{4}D_{\xi}^{4}\right)+\frac{7}{2025}h_{1}^{4}h_{2}^{4}h_{3}^{4}D_{\xi}^{4}D_{\eta}^{4}D_{\xi}^{4}\right].\end{gathered}

Other numerical integration formulas were given in [4].

§. 4. The Cavalieri-Simpson numerical integration formula in $E_{s}$

20.

In conclusion, we will give, in explicit form, two of the most important cubature formulas already deduced in the cases $s=1,2$ and 3 .

Thus we have the cubature formula of degree of accuracy $(1,1,\ldots,1)$

\iint\ldots\int_{D}f(M)dM=2^{s}m_{1}\ldots m_{s}h_{1}\ldots h_{s}f\left(M_{0}\right)+\rho

where $D$ is the hyperparallelepiped

t_{0}^{i}-m_{i}h_{i}\leqq t^{i}\leqq t_{0}^{i}+m_{i}h_{i}\quad(i=\overline{1,s})

and the rest has the expression

		$\displaystyle\rho=\frac{m_{1}m_{2}\ldots m_{s}h_{1}\ldots h_{s}}{3}\left[2^{s-1}m_{1}^{2}h_{1}^{2}D_{\xi_{1}}^{2}+\cdots+2^{s-1}m_{s}^{2}h_{s}^{2}D_{\xi_{s}}^{2}-2^{s-2}\frac{m_{1}^{2}m_{2}^{2}h_{1}^{2}h_{2}^{2}}{3}D_{\xi_{1}}^{2}D_{\xi_{2}}^{2}-\right.$
		$\displaystyle\quad\quad\ldots-2^{s-2}\frac{m_{s-1}^{2}m_{s}^{2}h_{s-1}^{2}h_{s}}{3}D_{\xi_{s-1}}^{2}D_{\xi_{s}}^{2}$
		$\displaystyle\left.+\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots h_{s}^{2}D_{\xi_{1}}^{2}\ldots D_{\xi_{3}}^{2}\right].$

21.

If in formula (39) it is made

p_{1}=p_{2}=\ldots=p_{s}=m_{1}=m_{2}=\ldots=m_{s}=1

the following cubature formula is obtained, with partial accuracy $(3,3,\ldots,3)$ , which represents the extension of the Cavalieri-Sompson formula in $E_{s}$

CONTRIBUTIONS TO L'INTÉGRATION NUMÉRIQUE DES FUNCTIONS DE PLUSIEURS VARIABLES

(Résumé)
By using certain interpolation formulas for the functions of several variables, several formulas have been constructed for the approximate calculation of defined multiple integrals. Pour chaque formule donnée on estabilit l'expression du reste.

Dans le premier paragraphe, après quelques considérations générales sur l'intégration numérique des fonctions de plusieurs variables, on déduit, en particulier, la formulae de Cavalieri-Simpson pour deux variables. A cette occasion on donne aussi une précis expression du reste (25) de cette formulae.

Dans le second paragraphe est constructed a cubature formula (39) pour les intégrales s-uples. Au (42) on établissement l'expression du reste de cette formulae.

Dans le troisième paragraphe on déduit sous une forme explicite, de (39), une série de formules d'intégration numérique poúr les integrales simple, double et triple.

Dans le dernier paragraphe on donne efficaciously two formulae de cubature pour les integrales s-uples: la formulae (62) which uses a single node and la formulae (64) which represents the generalization of the quadrature formulas of Cavalieri-Simpson.

Contributions to the numerical integration of functions of several variables

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

CONTRIBUTIONS TO THE NUMERICAL INTEGRATION OF FUNCTIONS OF MULTIPLE VARIABLES

§ 1. General considerations

§ 2. Some practical cubature formulas for integrals $s$ -uple

§ 3. Important particular cases of the previous cubature formulas

§. 4. The Cavalieri-Simpson numerical integration formula in $E_{s}$

CONTRIBUTIONS TO L'INTÉGRATION NUMÉRIQUE DES FUNCTIONS DE PLUSIEURS VARIABLES

Related Posts

Contributions to the numerical integration of functions of several variables

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

CONTRIBUTIONS TO THE NUMERICAL INTEGRATION OF FUNCTIONS OF MULTIPLE VARIABLES

§ 1. General considerations

§ 2. Some practical cubature formulas for integralsSs-uple

§ 3. Important particular cases of the previous cubature formulas

§. 4. The Cavalieri-Simpson numerical integration formula inIt isSE_{s}

CONTRIBUTIONS TO L'INTÉGRATION NUMÉRIQUE DES FUNCTIONS DE PLUSIEURS VARIABLES

Related Posts

§ 2. Some practical cubature formulas for integrals $s$ -uple

§. 4. The Cavalieri-Simpson numerical integration formula in $E_{s}$