10. Given that $y=f(x)$ is a continuous function on the interval $[0,1]$, and always $0 \leqslant f(x) \leqslant 1$, the integral $\int_{0}^{1} f(x) \mathrm{d} x$ can be approximately calculated using a random simulation method. First, generate two sets (each set containing $N$ numbers) of uniformly distributed random numbers $x_{1}$, $x_{2}, \cdots, x_{N}$ and $y_{1}, y_{2}, \cdots, y_{N}$ on the interval $[0,1]$, thereby obtaining $N$ points $\left(x_{i}, y_{i}\right)(i=1,2, \cdots, N)$; then count the number of points $N_{1}$ that satisfy $y_{i} \leqslant f\left(x_{i}\right)(i=1,2, \cdots, N)$. Thus, the approximate value of the integral $\int_{0}^{1} f(x) \mathrm{d} x$ obtained by the random simulation method is . $\qquad$
10. $\frac{N_{1}}{N}$.

Let $\int_{0}^{1} f(x) \mathrm{d} x=S$.
If $N$ points are taken uniformly at random in the region $x \in [0,1]$, $y \in [0,1]$ with an area of 1, and the number of points within the integration region is $N_{1}$, then $\frac{S}{1}=\frac{N_{1}}{N}$.
Therefore, the approximate value of $S$ is $\frac{N_{1}}{N}$.