Answer
$$\frac{13\pi }{3}$$
Work Step by Step
Given
$$y=\sqrt{5-x}, \ \ \ \ \ \ \ 3 \leqslant x \leqslant 5$$
Since
$$\frac{dy}{dx}= \frac{1}{2 \sqrt{5-x}}$$
Then
\begin{aligned}
S&= 2 \pi \int_{3}^{5} y\sqrt{1+\left[\frac{dy}{d x}\right]^{2}} d x\\
&=2 \pi \int_{3}^{5} \sqrt{5-x} \sqrt{1+\left(\frac{1}{2 \sqrt{5-x}}\right)^{2}}\\
&= 2 \pi \int_{3}^{5} \sqrt{5-x} \sqrt{\left(\frac{21-4 x}{4(5-x)}\right)} d x\\
&=2 \pi \int_{3}^{5} \sqrt{5-x}\left(\frac{\sqrt{21-4 x}}{2 \sqrt{5-x}}\right) d x\\
&=\pi \int_{3}^{5} \sqrt{21-4 x} d x\\
&=\frac{-\pi}{4}\frac{2}{3}(21-4x)^{3/2}\bigg|_{3}^{5}\\
&=\frac{-\pi}{6}[1-9^{3/2}]\\
&=\frac{13\pi }{3}
\end{aligned}
