Example 2. Approximately calculate the increase in the volume of a cylinder with height $H=40 \mathrm{~cm}$ and base radius $R=30 \mathrm{~cm}$ when the radius is increased by $0.5 \mathrm{~cm}$.
Solution. The volume of a cylinder $V$ with a constant height $H$ and a variable radius $x$ is a function of $x: V=\pi H x^{2}$. For $x=R=30$, we have $V=36000 \pi$. The increment of the cylinder's volume is replaced by its differential:

$$
\Delta V \approx d V=2 \pi H x \cdot \Delta x=2 \pi \cdot 40 \cdot 30 \cdot 0.5=1200 \pi
$$

Answer. $\Delta V=1200 \pi \mathrm{cm}^{3}$, new volume $37200 \pi \mathrm{cm}^{3}$.