748. Derive the formula $l_{3}^{*}=(b-a) c\left(n_{1} / n\right)$ for estimating the definite integral $I=\int_{a}^{b} \varphi(x) \mathrm{d} x$, where the integrand is non-negative and bounded ( $0 \leqslant \varphi(x) \leqslant c$ ), based on the interpretation of the integral as an area.
The problem is solved. Let's consider a two-dimensional random variable ( $X, Y$ ), uniformly distributed in a rectangle $D$ with base ( $b-a$ ) and height $c$, with probability density $f(x, y)=1 /(b-a) c$. The component $X$ is uniformly distributed in the interval ( $a, b$ ) with density $1 /(b-a)$; the component $Y$ is distributed in the interval ( 0, c) with density $1 / c$.

If $n$ points ( $x_{i}, y_{i}$ ) are generated, belonging to the rectangle $D$, of which $n_{i}$ points fall below the curve $y=\varphi(x)$, then the ratio of the area determined by the integral 1 to the area of the rectangle $D$

$$
\frac{\int_{a}^{b} \varphi(x) d x}{(b-a) c} \simeq \frac{n_{\mathbf{1}}}{n}
$$

From this,

$$
\int_{a}^{b} \varphi(x) \mathrm{d} x=(b-a) c\left(n_{1} n\right)
$$

Thus, as an estimate of the integral /, we can take

$$
l_{3}^{*}=(b-a) c\left(n_{1} / n\right)
$$