A surface S is obtained by rotation, around the axis OZ, the arc of parabola $\displaystyle \alpha : z = \frac{x^2}{2}, y = 0$ with $\displaystyle 0 \leq x \leq 1$. Determine the area of the surface obtained
Or you can recognize that the rotation will produce an elliptical paraboloid $\displaystyle z=\frac{x^2+y^2}{2}$ which can be parametrized as $\displaystyle \sigma(u,v)=\left(u,v, \frac{u^2+v^2}{2}\right)$
A switch to polar coordinates will then basically produce the same integral as Danny showed you.
$\displaystyle \iint_D \left| \vec {\sigma'_u}\times\vec {\sigma'_v}\right|dudv=\{\text{Polar coordinates}\}=\int \int_0^{2\pi}\sqrt{r^2+1} \cdot rd\varphi dr=$ $\displaystyle 2\pi\int r\sqrt{r^2+1}dr$