Example 2.60. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+(y-a)^{2}=R^{2}$ around the $O Y$ axis over the segment $0<y_{1} \leqslant y \leqslant y_{2}<R$.
Solution. Let's find the derivative of the implicit function

$$
2 x x_{y}^{\prime}+2(y-a)=0
$$

We will use the formula (2.31) $S=2 \pi \int_{y_{1}}^{y_{2}} x \sqrt{1+\left(x_{y}^{\prime}\right)^{2}} d y$ (bring $x$ under the square root)

$$
\begin{gathered}
2 \pi \int_{y_{1}}^{y_{2}} \sqrt{x^{2}+\left(x x_{y}^{\prime}\right)^{2}} d y=2 \pi \int_{y_{1}}^{y_{2}} \sqrt{R^{2}-(y-a)^{2}+(y-a)^{2}} d y= \\
=2 \pi R \int_{y_{1}}^{y_{2}} d y=2 \pi R\left(y_{2}-y_{1}\right)
\end{gathered}
$$