743. Find the estimate $I^{*}$ of the integral $I=\int_{0}^{1} \mathrm{~d} x \int_{k}^{1}(x+y) \mathrm{d} y$. Conduct 10 trials.
Solution. The region of integration is bounded by the lines $g=x, y=1, x=0$ and, obviously, belongs to the unit square. The area of the region of integration (a right triangle) $S=(1 \cdot 1) / 2=0.5$.

We use the formula

$$
f^{*}=S \cdot \frac{\sum_{i=1}^{N} f\left(x_{i}, y_{i}\right)}{N}=0.5 \frac{\sum_{i=1}^{N} f\left(x_{i}, g_{i}\right)}{N}
$$

where $N$ is the number of random points $(x_{i}, y_{i})$ that belong to the region of integration; for these points $y_{i} \geqslant x_{i}$ (for each trial where this condition is met, one is recorded in the counter $\boldsymbol{N}$). Pairs of independent random numbers $(x_{i}, y_{i})$ are taken from the table in Appendix 9, starting from the first row at the top. The results are as follows:

| Number of trial $i$ | $x_{i}$ | $y_{i}$ | Condition $y_{i}>x_{i}$ | $f\left(x_{i}, y_{i}\right)=x_{i}+y_{i}$ |
| :---: | :---: | :---: | :---: | :---: |
| 1  2  3  4  5  6  7  8  8  9  10 | 0.100  0.253  0.520  0.863  0.354  0.809  0.911  0.542  0.056  0.474 | 0.973  0.376  0.135  0.467  0.876  0.590  0.737  0.048  0.489  0.296 | 1  1  1 | 1.073  0.629  1.230    0.545 |
| $\mathbf{\Sigma}$ |  |  | 4 | 3.477 |

10 trials are presented in Table 65.

Table 65

From Table 65, we find $N=4, \sum_{i=1}^{4} f\left(x_{i}, y_{i}\right)=3.477$. Substituting these numbers into the formula (*), we obtain the desired estimate:

$$
I^{\circ}=0.5 \cdot(3.477 / 4)=0.435
$$

The relatively large discrepancy between the obtained estimate and the exact value $I=0.5$ is explained by the small number of trials.