Example 1. Using the formulas for rectangles, taking $n=10$, calculate $J=\int_{0}^{1} \frac{d x}{1+x^{3}}$.
Solution. Here $h=\frac{1-0}{10}=0.1, f(x)=\frac{1}{1+x^{3}}$. We will construct a table of the corresponding values of the argument and function necessary for the approximate calculation of the given integral (Table 13.1).

Table 13.1

| $\mathrm{k}$ | $x_{k}$ | $x_{k}^{3}$ | $1+x_{k}^{3}$ | $y_{k}=f\left(x_{k}\right)=$  $=\frac{1}{1+x_{k}^{3}}$ |
| :---: | :---: | :---: | :---: | :---: |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0.1 | 0.001 | 1.001 | 0.99900 |
| 2 | 0.2 | 0.008 | 1.008 | 0.99206 |
| 3 | 0.3 | 0.027 | 1.027 | 0.97371 |
| 4 | 0.4 | 0.064 | 1.064 | 0.93985 |
| 5 | 0.5 | 0.125 | 1.125 | 0.88889 |
| 6 | 0.6 | 0.216 | 1.216 | 0.82237 |
| 7 | 0.7 | 0.343 | 1.343 | 0.74460 |
| 8 | 0.8 | 0.512 | 1.512 | 0.66138 |
| 9 | 0.9 | 0.729 | 1.729 | 0.57837 |
| 10 | 1 | 1 | 2 | 0.5 |
|  |  |  | $\sum_{k=1}^{2} y_{k}=7.60023$ |  |

Since $\sum_{k=0}^{9} y_{k}=8.60023, \sum_{k=1}^{10} y_{k}=8.10023$, then by formulas (13.1) and

(13.2) respectively, we get: $J_{n}=0.1 \cdot 8.60023=0.86002$, $J_{n}=0.1 \cdot 8.10023=0.81002$.