Answer
$\frac{\pi}{24}(65\sqrt{65} - 17\sqrt{17})$
Work Step by Step
$x = 1+2y^{2}$
and
$1+(dx/dy)^{2} = 1+(4y)^{2} = 1 + 16y^{2}$
$S = 2\pi \int^{2}_{1} y \sqrt{1+16y^{2}} dy $
$= \frac{\pi}{16} \int^{2}_{1} (16y^{2}+1)^{1/2} 32y dy $
$= \frac{\pi}{24}(65\sqrt{65} - 17\sqrt{17})$
