A vector calculus problem.. 
I am a beginner in vector calculus . It will be great if someone can guide me solve this problem . Thanks. I don't know how to proceed. I am studying by online sources only so I have no teacher it will be great if someone solves this problem for me

First parametrize your surface $S$. One way to parametrize is given by spherical coordinates:
$$\phi:[0,\frac{\pi}{2}]\times[0,\frac{\pi}{2}]\to\mathbb{R}^3,\quad \phi(s,t)=(\cos s\cos t, \cos s\sin t,\sin s).$$
Here the domain is $[0,\frac{\pi}{2}]\times[0,\frac{\pi}{2}]$ because we want all three coordinates of $\phi(s,t)$ to be non-negative, as $S$ lies in the first octant. 
In this case we have $x=x(s,t)=\cos s\cos t.$ Then find the cross product of the partial derivatives of $\phi$:
$$\frac{\partial\phi}{\partial s}\times\frac{\partial\phi}{\partial t},$$
Then by definition of surface integral
$$\int x dS=\int_{[0,\frac{\pi}{2}]\times[0,\frac{\pi}{2}]}x(s,t)\left\|\frac{\partial\phi}{\partial s}\times\frac{\partial\phi}{\partial t}\right\|dsdt.$$
I left the detailed calculations for you to fill in.
