If we for example set $n=1$, we obtain $$\begin{equation*} P_1(x) = y_0\frac{x-x_1}{x_0-x_1}+y_1\frac{x-x_0}{x_1-x_0}=\frac{y_1-y_0}{x_1-x_0}x-\frac{y_1x_0+y_0x_1}{x_1-x_0}, \end{equation*}$$ which we recognize as the equation for a straight line.

The polynomial interpolatory quadrature of order $n$ with equidistant quadrature points $x_k=a+kh$ and step $h=(b-a)/n$ is called the Newton-Cotes quadrature formula of order $n$.