Example 2. Find the mass of the curve $y=\frac{1}{2} x^{2}+1$, if its linear density is $\rho(x)=x+1, x \in[0 ; \sqrt{3}]$.
Solution. We have $y=\frac{1}{2} x^{2}+1, \quad y^{\prime}=x, \quad d l=\sqrt{1+y^{\prime 2}} d x=$ $=\sqrt{1+x^{2}} d x$. Further,

$$
\begin{aligned}
m=\int_{l} \rho(x) d l= & \int_{0}^{\sqrt{3}}(x+1) \sqrt{1+x^{2}} d x= \\
& =\int_{0}^{\sqrt{3}} x \sqrt{1+x^{2}} d x+\int_{0}^{\sqrt{3}} \sqrt{1+x^{2}} d x=
\end{aligned}
$$

K.N. Lungu, E.V. Makarov

$$
\begin{aligned}
=\left.\frac{1}{3} \sqrt{\left(1+x^{2}\right)^{3}}\right|_{0} ^{\sqrt{3}}+\frac{1}{2}\left(x \sqrt{1+x^{2}}\right. & \left.+\ln \left(x+\sqrt{1+x^{2}}\right)\right)\left.\right|_{0} ^{\sqrt{3}}= \\
& =\frac{7}{3}+\frac{1}{2}(2 \sqrt{3}+\ln (2+\sqrt{3}))
\end{aligned}
$$

Answer. $m=\frac{7}{3}+\frac{1}{2}(\ln (2+\sqrt{3})+2 \sqrt{3})$.