Example 3. Calculate the mass $m$ and the moment of inertia of a uniform flat rod of length $l$ relative to its end (density is $\rho$).
Solution. Let's align the rod with the segment $[0 ; l]$ of the $O x$ axis (the rod is defined by the graph of the function $y=0, x \in[0 ; l]$. Then $\left(\sqrt{1+y^{\prime 2}}=1\right)$

$$
m=\int_{0}^{l} \rho d x=\rho l
$$

The moment of inertia of the rod relative to its end is

$$
M_{y}=\int_{0}^{l} \rho x^{2} d x=\frac{\rho l^{3}}{3}
$$

Answer. $\rho l ; \frac{\rho l^{3}}{3}$.