Surface Integral without Parametrization 
Let $S$ be the sphere $x^2 + y^2 + z^2 = a^2$.
  
  
*
  
*Use symmetry considerations to evaluate $\iint_Sx\,dS$ without resorting to parametrizing the sphere.
  
*Let F $= (1, 1, 1)$. Use symmetry to determine the vector surface integral $\iint_S$F$\cdot \, dS$ without parametrizing
  the sphere.
  

I find it difficult to understand question 1. What does it mean by 'symmetry considerations'? 
For question 2, without parametrizing the sphere, I find out that the integral should be equal to $$\frac1a \iint_S(x+y+z)dS.$$ Now symmetry comes in - a sphere is symmetric in all axes, thus the answer should be $\frac3a$ times the answer in part a. Am I correct?
Any help will be appreciated.

Integrals of the form $$\iint x \, dS$$ refer to center of mass calculations. This double integral will give the $x$ coordinate of the center of mass of your surface, likewise for $y$ and $z$. However, the center of mass of a sphere is at its center, and this sphere is centered at the origin. Therefore, this surface integral is zero.
For question $2$, note that a normal vector is $(x,y,z)$. What happens when you compute ${\mathbf F} \cdot (x,y,z)$?
