According to Newton's law of universal gravitation, a material point $P$ with mass $m$ attracts a material point $P_0$ with mass $m_0$ with a force directed from $P_0$ towards $P$, of size $k\cfrac{mm_0}{r^2}$ (further for simplicity k=1).
When point $P_0$ is attracted by $n$ points $P_1, P_2,\ldots, P_n$ with masses $m_1,m_2,\ldots,m_n$, to obtain the resultant force, the attraction forces of individual points must be added geometrically. If we denote the projections of the resultant on the axes by $X$ and $Y$, and the angle made by the vector $r_i=\overline{P_0P_i}$ with the $x$ axis by $\alpha_i$, we get $$X=\sum_{i=1}^{n}\frac{m_0m_i}{r_i^2}\cos{\alpha_i}, \hspace{1cm} Y=\sum_{i=1}^{n} \frac{m_0m_i}{r_i^2}\sin{\alpha_i},$$ where $r_i$ is the length of the vector $r_i$.
Now let us establish that the attractive mass has a continuous distribution on the $\sigma$ curve. Then let's divide the curve into parts and find the approximate values of the projections of the resultant on the axes. Concentrating the mass of each part at any chosen point $P_i$ and taking into account that the mass of the $i$-th part is approximately equal to $f(P_i)\cdot\Delta s_i$, we obtain $$X\approx\sum_{i}\frac{m_0f(P_i)\Delta s_i}{r_i^2}\cos{\alpha_i}, \hspace{1cm} Y\approx\sum_{i}\frac{m_0f (P_i)\Delta s_i}{r_i^2}\sin{\alpha_i},$$.
So far everything is clear to me. Unfortunately, I don't understand the next transition. How to explain this?
If all $\Delta s_i$ tend to zero, we get $$X=m_0\int_{\sigma} \frac{f(P)\cos{\alpha}}{r^2} ds, \hspace{1cm} Y=m_0\int_{\sigma} \frac{f (P)\sin{\alpha}}{r^2} ds;$$
where $r$ is the length of the vector $r=\overline{P_0P}$, and $\alpha$ - the angle that this vector makes with the $x$ axis.
This transition from the sum to the integral is essentially an application of the limiting process inherent to the definition of the Riemann integral. I'll break it down for you:
Discretization and Summation: First, note that when you're approximating the forces from each segment of the curve, you're effectively discretizing the curve. The curve $ \sigma $ is divided into small segments, each of length $ \Delta s_i $, and the force due to each segment is computed using the approximate mass $ f(P_i) \cdot \Delta s_i $ located at a chosen point $ P_i $ within that segment.
Limiting Process and Integration: The Riemann integral is, conceptually, the limit of the sum as the size of the segments tends to zero. That is, the curve is divided into smaller and smaller segments, making the approximation more accurate. If you let all $ \Delta s_i $ tend to zero (which means the number of segments is going to infinity, and the length of each segment is becoming infinitesimally small), then these approximating sums: $$ \begin{align} X & \approx \sum_{i} \frac{m_0 f(P_i) \Delta s_i}{r_i^2} \cos{\alpha_i} \\ Y &\approx \sum_{i} \frac{m_0 f(P_i) \Delta s_i}{r_i^2} \sin{\alpha_i} \end{align} $$ become the exact integrals: $$ \begin{align} X &= m_0 \int_{\sigma} \frac{f(P) \cos{\alpha}}{r^2} ds \\ Y &= m_0 \int_{\sigma} \frac{f(P) \sin{\alpha}}{r^2} ds \end{align} $$
Geometric Interpretation: Think of the curve $\sigma$ as being made up of an infinite number of infinitesimally small segments. Each segment exerts a tiny force on the point $P_0$. The net force due to the entire curve is the sum of the forces due to all these tiny segments. This summation, in the limit as each segment size approaches zero, is represented by the integral.
In summary, the transition from the summation to the integral is a way of saying: "Instead of approximating the force due to each segment of the curve, let's compute the exact force by considering all the infinitesimally small pieces of the curve". This is the essence of integration, and this transition is a fundamental concept in calculus, often visualized in the move from Riemann sums to definite integrals.