Answer
$\dfrac{16 \sqrt 2-5 \sqrt 5 }{12}\pi $
Work Step by Step
The formula to determine the surface area is as follows:
$S= \int_{m}^{n} 2 \pi y \sqrt {1+(\dfrac{dy}{dx})^2}$
Now, $S=\int_{5/8}^{1} 2 \pi ( \sqrt {2y-1}) \sqrt {\dfrac{2y}{2y-1} } dy\\= 2 \sqrt 2 \pi \times \int_{(5/8)}^{1} (\dfrac{2}{3}) y^{3/2} dy \\ =\dfrac{ 4 \sqrt 2 \pi}{3} (1- \dfrac{5 \sqrt 5} {16 \sqrt 2}) \\=\dfrac{(16 \sqrt 2-5 \sqrt 5) \pi }{12}$
